# Valid Cliques, Independent Sets, Inclusion-Maximality and Global Extrema

This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.

This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.

batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations · 2026-10-14 · 100 questions

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

Field order uses a source-bound presentation supplement; canonical JSONL and mathematical answers are unchanged.

[Field-order and source-binding data](https://ocme.evemisslab.com/data/i18n/question-output-protocols.json)



## 1. Only a Valid Clique Size Certifies a Clique-Number Lower Bound

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [1, 2], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[0, 1, 2], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_clique, candidate_size, boundary).

View answer and explanation([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega)

Check adjacency of every distinct selected pair. The reported candidate size gives a lower bound on omega only if the validity flag is Yes; it is not by itself a maximum certificate or an exact chromatic number. The exact result, retaining the requested order, is ([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0001/)

Cliques and independent sets · qf-mod37-clique-clique-audit · candidate





## 2. Only a Valid Clique Size Certifies a Clique-Number Lower Bound

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [1, 2], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[0, 1, 2], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_clique, candidate_size, boundary).

View answer and explanation([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega)

Check adjacency of every distinct selected pair. The reported candidate size gives a lower bound on omega only if the validity flag is Yes; it is not by itself a maximum certificate or an exact chromatic number. The exact result, retaining the requested order, is ([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0002/)

Cliques and independent sets · qf-mod37-clique-clique-audit · candidate





## 3. Only a Valid Clique Size Certifies a Clique-Number Lower Bound

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [1, 2], [1, 3], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[0, 1, 2], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_clique, candidate_size, boundary).

View answer and explanation([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega)

Check adjacency of every distinct selected pair. The reported candidate size gives a lower bound on omega only if the validity flag is Yes; it is not by itself a maximum certificate or an exact chromatic number. The exact result, retaining the requested order, is ([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0003/)

Cliques and independent sets · qf-mod37-clique-clique-audit · candidate





## 4. Only a Valid Clique Size Certifies a Clique-Number Lower Bound

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[0, 1, 2], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_clique, candidate_size, boundary).

View answer and explanation([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega)

Check adjacency of every distinct selected pair. The reported candidate size gives a lower bound on omega only if the validity flag is Yes; it is not by itself a maximum certificate or an exact chromatic number. The exact result, retaining the requested order, is ([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0004/)

Cliques and independent sets · qf-mod37-clique-clique-audit · candidate





## 5. Only a Valid Clique Size Certifies a Clique-Number Lower Bound

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [1, 2], [2, 4], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[0, 1, 2], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_clique, candidate_size, boundary).

View answer and explanation([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega)

Check adjacency of every distinct selected pair. The reported candidate size gives a lower bound on omega only if the validity flag is Yes; it is not by itself a maximum certificate or an exact chromatic number. The exact result, retaining the requested order, is ([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0005/)

Cliques and independent sets · qf-mod37-clique-clique-audit · candidate





## 6. Only a Valid Clique Size Certifies a Clique-Number Lower Bound

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [1, 2], [2, 4], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[0, 1, 2], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_clique, candidate_size, boundary).

View answer and explanation([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega)

Check adjacency of every distinct selected pair. The reported candidate size gives a lower bound on omega only if the validity flag is Yes; it is not by itself a maximum certificate or an exact chromatic number. The exact result, retaining the requested order, is ([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0006/)

Cliques and independent sets · qf-mod37-clique-clique-audit · candidate





## 7. Only a Valid Clique Size Certifies a Clique-Number Lower Bound

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [1, 2], [1, 3], [2, 4], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[0, 1, 2], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_clique, candidate_size, boundary).

View answer and explanation([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega)

Check adjacency of every distinct selected pair. The reported candidate size gives a lower bound on omega only if the validity flag is Yes; it is not by itself a maximum certificate or an exact chromatic number. The exact result, retaining the requested order, is ([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0007/)

Cliques and independent sets · qf-mod37-clique-clique-audit · candidate





## 8. Only a Valid Clique Size Certifies a Clique-Number Lower Bound

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [2, 4], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[0, 1, 2], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_clique, candidate_size, boundary).

View answer and explanation([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega)

Check adjacency of every distinct selected pair. The reported candidate size gives a lower bound on omega only if the validity flag is Yes; it is not by itself a maximum certificate or an exact chromatic number. The exact result, retaining the requested order, is ([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0008/)

Cliques and independent sets · qf-mod37-clique-clique-audit · candidate





## 9. Only a Valid Clique Size Certifies a Clique-Number Lower Bound

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 5], [1, 2], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[0, 1, 2], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_clique, candidate_size, boundary).

View answer and explanation([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega)

Check adjacency of every distinct selected pair. The reported candidate size gives a lower bound on omega only if the validity flag is Yes; it is not by itself a maximum certificate or an exact chromatic number. The exact result, retaining the requested order, is ([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0009/)

Cliques and independent sets · qf-mod37-clique-clique-audit · candidate





## 10. Only a Valid Clique Size Certifies a Clique-Number Lower Bound

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [0, 5], [1, 2], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[0, 1, 2], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_clique, candidate_size, boundary).

View answer and explanation([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega)

Check adjacency of every distinct selected pair. The reported candidate size gives a lower bound on omega only if the validity flag is Yes; it is not by itself a maximum certificate or an exact chromatic number. The exact result, retaining the requested order, is ([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0010/)

Cliques and independent sets · qf-mod37-clique-clique-audit · candidate





## 11. Only a Valid Clique Size Certifies a Clique-Number Lower Bound

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 5], [1, 2], [1, 3], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[0, 1, 2], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_clique, candidate_size, boundary).

View answer and explanation([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega)

Check adjacency of every distinct selected pair. The reported candidate size gives a lower bound on omega only if the validity flag is Yes; it is not by itself a maximum certificate or an exact chromatic number. The exact result, retaining the requested order, is ([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0011/)

Cliques and independent sets · qf-mod37-clique-clique-audit · candidate





## 12. Only a Valid Clique Size Certifies a Clique-Number Lower Bound

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [0, 5], [1, 2], [1, 3], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[0, 1, 2], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_clique, candidate_size, boundary).

View answer and explanation([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega)

Check adjacency of every distinct selected pair. The reported candidate size gives a lower bound on omega only if the validity flag is Yes; it is not by itself a maximum certificate or an exact chromatic number. The exact result, retaining the requested order, is ([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0012/)

Cliques and independent sets · qf-mod37-clique-clique-audit · candidate





## 13. Only a Valid Clique Size Certifies a Clique-Number Lower Bound

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 5], [1, 2], [2, 4], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[0, 1, 2], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_clique, candidate_size, boundary).

View answer and explanation([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega)

Check adjacency of every distinct selected pair. The reported candidate size gives a lower bound on omega only if the validity flag is Yes; it is not by itself a maximum certificate or an exact chromatic number. The exact result, retaining the requested order, is ([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0013/)

Cliques and independent sets · qf-mod37-clique-clique-audit · candidate





## 14. Only a Valid Clique Size Certifies a Clique-Number Lower Bound

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [0, 5], [1, 2], [2, 4], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[0, 1, 2], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_clique, candidate_size, boundary).

View answer and explanation([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega)

Check adjacency of every distinct selected pair. The reported candidate size gives a lower bound on omega only if the validity flag is Yes; it is not by itself a maximum certificate or an exact chromatic number. The exact result, retaining the requested order, is ([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0014/)

Cliques and independent sets · qf-mod37-clique-clique-audit · candidate





## 15. Only a Valid Clique Size Certifies a Clique-Number Lower Bound

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 5], [1, 2], [1, 3], [2, 4], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[0, 1, 2], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_clique, candidate_size, boundary).

View answer and explanation([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega)

Check adjacency of every distinct selected pair. The reported candidate size gives a lower bound on omega only if the validity flag is Yes; it is not by itself a maximum certificate or an exact chromatic number. The exact result, retaining the requested order, is ([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0015/)

Cliques and independent sets · qf-mod37-clique-clique-audit · candidate





## 16. Only a Valid Clique Size Certifies a Clique-Number Lower Bound

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [0, 5], [1, 2], [1, 3], [2, 4], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[0, 1, 2], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_clique, candidate_size, boundary).

View answer and explanation([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega)

Check adjacency of every distinct selected pair. The reported candidate size gives a lower bound on omega only if the validity flag is Yes; it is not by itself a maximum certificate or an exact chromatic number. The exact result, retaining the requested order, is ([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0016/)

Cliques and independent sets · qf-mod37-clique-clique-audit · candidate





## 17. Only a Valid Clique Size Certifies a Clique-Number Lower Bound

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [1, 2], [1, 5], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[0, 1, 2], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_clique, candidate_size, boundary).

View answer and explanation([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega)

Check adjacency of every distinct selected pair. The reported candidate size gives a lower bound on omega only if the validity flag is Yes; it is not by itself a maximum certificate or an exact chromatic number. The exact result, retaining the requested order, is ([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0017/)

Cliques and independent sets · qf-mod37-clique-clique-audit · candidate





## 18. Only a Valid Clique Size Certifies a Clique-Number Lower Bound

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 5], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[0, 1, 2], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_clique, candidate_size, boundary).

View answer and explanation([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega)

Check adjacency of every distinct selected pair. The reported candidate size gives a lower bound on omega only if the validity flag is Yes; it is not by itself a maximum certificate or an exact chromatic number. The exact result, retaining the requested order, is ([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0018/)

Cliques and independent sets · qf-mod37-clique-clique-audit · candidate





## 19. Only a Valid Clique Size Certifies a Clique-Number Lower Bound

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [1, 2], [1, 3], [1, 5], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[0, 1, 2], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_clique, candidate_size, boundary).

View answer and explanation([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega)

Check adjacency of every distinct selected pair. The reported candidate size gives a lower bound on omega only if the validity flag is Yes; it is not by itself a maximum certificate or an exact chromatic number. The exact result, retaining the requested order, is ([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0019/)

Cliques and independent sets · qf-mod37-clique-clique-audit · candidate





## 20. Only a Valid Clique Size Certifies a Clique-Number Lower Bound

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [1, 5], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[0, 1, 2], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_clique, candidate_size, boundary).

View answer and explanation([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega)

Check adjacency of every distinct selected pair. The reported candidate size gives a lower bound on omega only if the validity flag is Yes; it is not by itself a maximum certificate or an exact chromatic number. The exact result, retaining the requested order, is ([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0020/)

Cliques and independent sets · qf-mod37-clique-clique-audit · candidate





## 21. Only a Valid Clique Size Certifies a Clique-Number Lower Bound

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [1, 2], [1, 5], [2, 4], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[0, 1, 2], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_clique, candidate_size, boundary).

View answer and explanation([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega)

Check adjacency of every distinct selected pair. The reported candidate size gives a lower bound on omega only if the validity flag is Yes; it is not by itself a maximum certificate or an exact chromatic number. The exact result, retaining the requested order, is ([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0021/)

Cliques and independent sets · qf-mod37-clique-clique-audit · candidate





## 22. Only a Valid Clique Size Certifies a Clique-Number Lower Bound

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 5], [2, 4], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[0, 1, 2], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_clique, candidate_size, boundary).

View answer and explanation([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega)

Check adjacency of every distinct selected pair. The reported candidate size gives a lower bound on omega only if the validity flag is Yes; it is not by itself a maximum certificate or an exact chromatic number. The exact result, retaining the requested order, is ([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0022/)

Cliques and independent sets · qf-mod37-clique-clique-audit · candidate





## 23. Only a Valid Clique Size Certifies a Clique-Number Lower Bound

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [1, 2], [1, 3], [1, 5], [2, 4], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[0, 1, 2], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_clique, candidate_size, boundary).

View answer and explanation([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega)

Check adjacency of every distinct selected pair. The reported candidate size gives a lower bound on omega only if the validity flag is Yes; it is not by itself a maximum certificate or an exact chromatic number. The exact result, retaining the requested order, is ([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0023/)

Cliques and independent sets · qf-mod37-clique-clique-audit · candidate





## 24. Only a Valid Clique Size Certifies a Clique-Number Lower Bound

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [1, 5], [2, 4], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[0, 1, 2], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_clique, candidate_size, boundary).

View answer and explanation([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega)

Check adjacency of every distinct selected pair. The reported candidate size gives a lower bound on omega only if the validity flag is Yes; it is not by itself a maximum certificate or an exact chromatic number. The exact result, retaining the requested order, is ([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0024/)

Cliques and independent sets · qf-mod37-clique-clique-audit · candidate





## 25. Only a Valid Clique Size Certifies a Clique-Number Lower Bound

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 5], [1, 2], [1, 5], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[0, 1, 2], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_clique, candidate_size, boundary).

View answer and explanation([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega)

Check adjacency of every distinct selected pair. The reported candidate size gives a lower bound on omega only if the validity flag is Yes; it is not by itself a maximum certificate or an exact chromatic number. The exact result, retaining the requested order, is ([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0025/)

Cliques and independent sets · qf-mod37-clique-clique-audit · candidate





## 26. An Independent Set Is a Clique in the Complement

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [1, 2], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 5], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_independent, clique_in_complement, candidate_size, boundary).

View answer and explanation([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha)

Check nonadjacency of every distinct selected pair and the corresponding complement-clique condition. A valid candidate size bounds alpha below, not above, and maximality/maximum need their own checks. The exact result, retaining the requested order, is ([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0026/)

Cliques and independent sets · qf-mod37-clique-independent-audit · candidate





## 27. An Independent Set Is a Clique in the Complement

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [1, 2], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 5], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_independent, clique_in_complement, candidate_size, boundary).

View answer and explanation([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha)

Check nonadjacency of every distinct selected pair and the corresponding complement-clique condition. A valid candidate size bounds alpha below, not above, and maximality/maximum need their own checks. The exact result, retaining the requested order, is ([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0027/)

Cliques and independent sets · qf-mod37-clique-independent-audit · candidate





## 28. An Independent Set Is a Clique in the Complement

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [1, 2], [1, 3], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 5], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_independent, clique_in_complement, candidate_size, boundary).

View answer and explanation([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha)

Check nonadjacency of every distinct selected pair and the corresponding complement-clique condition. A valid candidate size bounds alpha below, not above, and maximality/maximum need their own checks. The exact result, retaining the requested order, is ([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0028/)

Cliques and independent sets · qf-mod37-clique-independent-audit · candidate





## 29. An Independent Set Is a Clique in the Complement

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 5], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_independent, clique_in_complement, candidate_size, boundary).

View answer and explanation([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha)

Check nonadjacency of every distinct selected pair and the corresponding complement-clique condition. A valid candidate size bounds alpha below, not above, and maximality/maximum need their own checks. The exact result, retaining the requested order, is ([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0029/)

Cliques and independent sets · qf-mod37-clique-independent-audit · candidate





## 30. An Independent Set Is a Clique in the Complement

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [1, 2], [2, 4], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 5], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_independent, clique_in_complement, candidate_size, boundary).

View answer and explanation([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha)

Check nonadjacency of every distinct selected pair and the corresponding complement-clique condition. A valid candidate size bounds alpha below, not above, and maximality/maximum need their own checks. The exact result, retaining the requested order, is ([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0030/)

Cliques and independent sets · qf-mod37-clique-independent-audit · candidate





## 31. An Independent Set Is a Clique in the Complement

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [1, 2], [2, 4], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 5], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_independent, clique_in_complement, candidate_size, boundary).

View answer and explanation([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha)

Check nonadjacency of every distinct selected pair and the corresponding complement-clique condition. A valid candidate size bounds alpha below, not above, and maximality/maximum need their own checks. The exact result, retaining the requested order, is ([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0031/)

Cliques and independent sets · qf-mod37-clique-independent-audit · candidate





## 32. An Independent Set Is a Clique in the Complement

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [1, 2], [1, 3], [2, 4], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 5], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_independent, clique_in_complement, candidate_size, boundary).

View answer and explanation([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha)

Check nonadjacency of every distinct selected pair and the corresponding complement-clique condition. A valid candidate size bounds alpha below, not above, and maximality/maximum need their own checks. The exact result, retaining the requested order, is ([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0032/)

Cliques and independent sets · qf-mod37-clique-independent-audit · candidate





## 33. An Independent Set Is a Clique in the Complement

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [2, 4], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 5], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_independent, clique_in_complement, candidate_size, boundary).

View answer and explanation([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha)

Check nonadjacency of every distinct selected pair and the corresponding complement-clique condition. A valid candidate size bounds alpha below, not above, and maximality/maximum need their own checks. The exact result, retaining the requested order, is ([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0033/)

Cliques and independent sets · qf-mod37-clique-independent-audit · candidate





## 34. An Independent Set Is a Clique in the Complement

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 5], [1, 2], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 5], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_independent, clique_in_complement, candidate_size, boundary).

View answer and explanation([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha)

Check nonadjacency of every distinct selected pair and the corresponding complement-clique condition. A valid candidate size bounds alpha below, not above, and maximality/maximum need their own checks. The exact result, retaining the requested order, is ([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0034/)

Cliques and independent sets · qf-mod37-clique-independent-audit · candidate





## 35. An Independent Set Is a Clique in the Complement

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [0, 5], [1, 2], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 5], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_independent, clique_in_complement, candidate_size, boundary).

View answer and explanation([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha)

Check nonadjacency of every distinct selected pair and the corresponding complement-clique condition. A valid candidate size bounds alpha below, not above, and maximality/maximum need their own checks. The exact result, retaining the requested order, is ([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0035/)

Cliques and independent sets · qf-mod37-clique-independent-audit · candidate





## 36. An Independent Set Is a Clique in the Complement

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 5], [1, 2], [1, 3], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 5], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_independent, clique_in_complement, candidate_size, boundary).

View answer and explanation([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha)

Check nonadjacency of every distinct selected pair and the corresponding complement-clique condition. A valid candidate size bounds alpha below, not above, and maximality/maximum need their own checks. The exact result, retaining the requested order, is ([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0036/)

Cliques and independent sets · qf-mod37-clique-independent-audit · candidate





## 37. An Independent Set Is a Clique in the Complement

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [0, 5], [1, 2], [1, 3], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 5], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_independent, clique_in_complement, candidate_size, boundary).

View answer and explanation([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha)

Check nonadjacency of every distinct selected pair and the corresponding complement-clique condition. A valid candidate size bounds alpha below, not above, and maximality/maximum need their own checks. The exact result, retaining the requested order, is ([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0037/)

Cliques and independent sets · qf-mod37-clique-independent-audit · candidate





## 38. An Independent Set Is a Clique in the Complement

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 5], [1, 2], [2, 4], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 5], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_independent, clique_in_complement, candidate_size, boundary).

View answer and explanation([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha)

Check nonadjacency of every distinct selected pair and the corresponding complement-clique condition. A valid candidate size bounds alpha below, not above, and maximality/maximum need their own checks. The exact result, retaining the requested order, is ([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0038/)

Cliques and independent sets · qf-mod37-clique-independent-audit · candidate





## 39. An Independent Set Is a Clique in the Complement

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [0, 5], [1, 2], [2, 4], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 5], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_independent, clique_in_complement, candidate_size, boundary).

View answer and explanation([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha)

Check nonadjacency of every distinct selected pair and the corresponding complement-clique condition. A valid candidate size bounds alpha below, not above, and maximality/maximum need their own checks. The exact result, retaining the requested order, is ([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0039/)

Cliques and independent sets · qf-mod37-clique-independent-audit · candidate





## 40. An Independent Set Is a Clique in the Complement

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 5], [1, 2], [1, 3], [2, 4], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 5], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_independent, clique_in_complement, candidate_size, boundary).

View answer and explanation([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha)

Check nonadjacency of every distinct selected pair and the corresponding complement-clique condition. A valid candidate size bounds alpha below, not above, and maximality/maximum need their own checks. The exact result, retaining the requested order, is ([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0040/)

Cliques and independent sets · qf-mod37-clique-independent-audit · candidate





## 41. An Independent Set Is a Clique in the Complement

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [0, 5], [1, 2], [1, 3], [2, 4], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 5], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_independent, clique_in_complement, candidate_size, boundary).

View answer and explanation([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha)

Check nonadjacency of every distinct selected pair and the corresponding complement-clique condition. A valid candidate size bounds alpha below, not above, and maximality/maximum need their own checks. The exact result, retaining the requested order, is ([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0041/)

Cliques and independent sets · qf-mod37-clique-independent-audit · candidate





## 42. An Independent Set Is a Clique in the Complement

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [1, 2], [1, 5], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 5], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_independent, clique_in_complement, candidate_size, boundary).

View answer and explanation([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha)

Check nonadjacency of every distinct selected pair and the corresponding complement-clique condition. A valid candidate size bounds alpha below, not above, and maximality/maximum need their own checks. The exact result, retaining the requested order, is ([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0042/)

Cliques and independent sets · qf-mod37-clique-independent-audit · candidate





## 43. An Independent Set Is a Clique in the Complement

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 5], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 5], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_independent, clique_in_complement, candidate_size, boundary).

View answer and explanation([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha)

Check nonadjacency of every distinct selected pair and the corresponding complement-clique condition. A valid candidate size bounds alpha below, not above, and maximality/maximum need their own checks. The exact result, retaining the requested order, is ([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0043/)

Cliques and independent sets · qf-mod37-clique-independent-audit · candidate





## 44. An Independent Set Is a Clique in the Complement

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [1, 2], [1, 3], [1, 5], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 5], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_independent, clique_in_complement, candidate_size, boundary).

View answer and explanation([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha)

Check nonadjacency of every distinct selected pair and the corresponding complement-clique condition. A valid candidate size bounds alpha below, not above, and maximality/maximum need their own checks. The exact result, retaining the requested order, is ([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0044/)

Cliques and independent sets · qf-mod37-clique-independent-audit · candidate





## 45. An Independent Set Is a Clique in the Complement

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [1, 5], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 5], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_independent, clique_in_complement, candidate_size, boundary).

View answer and explanation([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha)

Check nonadjacency of every distinct selected pair and the corresponding complement-clique condition. A valid candidate size bounds alpha below, not above, and maximality/maximum need their own checks. The exact result, retaining the requested order, is ([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0045/)

Cliques and independent sets · qf-mod37-clique-independent-audit · candidate





## 46. An Independent Set Is a Clique in the Complement

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [1, 2], [1, 5], [2, 4], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 5], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_independent, clique_in_complement, candidate_size, boundary).

View answer and explanation([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha)

Check nonadjacency of every distinct selected pair and the corresponding complement-clique condition. A valid candidate size bounds alpha below, not above, and maximality/maximum need their own checks. The exact result, retaining the requested order, is ([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0046/)

Cliques and independent sets · qf-mod37-clique-independent-audit · candidate





## 47. An Independent Set Is a Clique in the Complement

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 5], [2, 4], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 5], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_independent, clique_in_complement, candidate_size, boundary).

View answer and explanation([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha)

Check nonadjacency of every distinct selected pair and the corresponding complement-clique condition. A valid candidate size bounds alpha below, not above, and maximality/maximum need their own checks. The exact result, retaining the requested order, is ([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0047/)

Cliques and independent sets · qf-mod37-clique-independent-audit · candidate





## 48. An Independent Set Is a Clique in the Complement

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [1, 2], [1, 3], [1, 5], [2, 4], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 5], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_independent, clique_in_complement, candidate_size, boundary).

View answer and explanation([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha)

Check nonadjacency of every distinct selected pair and the corresponding complement-clique condition. A valid candidate size bounds alpha below, not above, and maximality/maximum need their own checks. The exact result, retaining the requested order, is ([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0048/)

Cliques and independent sets · qf-mod37-clique-independent-audit · candidate





## 49. An Independent Set Is a Clique in the Complement

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [1, 5], [2, 4], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 5], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_independent, clique_in_complement, candidate_size, boundary).

View answer and explanation([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha)

Check nonadjacency of every distinct selected pair and the corresponding complement-clique condition. A valid candidate size bounds alpha below, not above, and maximality/maximum need their own checks. The exact result, retaining the requested order, is ([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0049/)

Cliques and independent sets · qf-mod37-clique-independent-audit · candidate





## 50. An Independent Set Is a Clique in the Complement

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 5], [1, 2], [1, 5], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 5], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_independent, clique_in_complement, candidate_size, boundary).

View answer and explanation([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha)

Check nonadjacency of every distinct selected pair and the corresponding complement-clique condition. A valid candidate size bounds alpha below, not above, and maximality/maximum need their own checks. The exact result, retaining the requested order, is ([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0050/)

Cliques and independent sets · qf-mod37-clique-independent-audit · candidate





## 51. Exact Extremal Sizes Need the Complete Vertex-Subset Search

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [1, 2], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (maximum_clique, clique_number, maximum_independent_set, independence_number, all_subsets_checked, certificate).

View answer and explanation([0, 1, 2], 3, [0, 3, 5], 3, Yes, exhaustive_all_vertex_subsets)

Inspect all vertex subsets. Return a maximum clique and independent set, preserving the first maximum witness in increasing vertex-bitmask order, not re-sorting tied witnesses lexicographically. This bounded exhaustive certificate is not a large-graph efficiency claim. The exact result, retaining the requested order, is ([0, 1, 2], 3, [0, 3, 5], 3, Yes, exhaustive_all_vertex_subsets).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0051/)

Cliques and independent sets · qf-mod37-clique-exact-extrema · candidate





## 52. Exact Extremal Sizes Need the Complete Vertex-Subset Search

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [1, 2], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (maximum_clique, clique_number, maximum_independent_set, independence_number, all_subsets_checked, certificate).

View answer and explanation([0, 1, 2], 3, [1, 3, 5], 3, Yes, exhaustive_all_vertex_subsets)

Inspect all vertex subsets. Return a maximum clique and independent set, preserving the first maximum witness in increasing vertex-bitmask order, not re-sorting tied witnesses lexicographically. This bounded exhaustive certificate is not a large-graph efficiency claim. The exact result, retaining the requested order, is ([0, 1, 2], 3, [1, 3, 5], 3, Yes, exhaustive_all_vertex_subsets).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0052/)

Cliques and independent sets · qf-mod37-clique-exact-extrema · candidate





## 53. Exact Extremal Sizes Need the Complete Vertex-Subset Search

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [1, 2], [1, 3], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (maximum_clique, clique_number, maximum_independent_set, independence_number, all_subsets_checked, certificate).

View answer and explanation([0, 1, 2], 3, [0, 3, 5], 3, Yes, exhaustive_all_vertex_subsets)

Inspect all vertex subsets. Return a maximum clique and independent set, preserving the first maximum witness in increasing vertex-bitmask order, not re-sorting tied witnesses lexicographically. This bounded exhaustive certificate is not a large-graph efficiency claim. The exact result, retaining the requested order, is ([0, 1, 2], 3, [0, 3, 5], 3, Yes, exhaustive_all_vertex_subsets).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0053/)

Cliques and independent sets · qf-mod37-clique-exact-extrema · candidate





## 54. Exact Extremal Sizes Need the Complete Vertex-Subset Search

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (maximum_clique, clique_number, maximum_independent_set, independence_number, all_subsets_checked, certificate).

View answer and explanation([0, 1, 2], 3, [2, 3, 5], 3, Yes, exhaustive_all_vertex_subsets)

Inspect all vertex subsets. Return a maximum clique and independent set, preserving the first maximum witness in increasing vertex-bitmask order, not re-sorting tied witnesses lexicographically. This bounded exhaustive certificate is not a large-graph efficiency claim. The exact result, retaining the requested order, is ([0, 1, 2], 3, [2, 3, 5], 3, Yes, exhaustive_all_vertex_subsets).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0054/)

Cliques and independent sets · qf-mod37-clique-exact-extrema · candidate





## 55. Exact Extremal Sizes Need the Complete Vertex-Subset Search

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [1, 2], [2, 4], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (maximum_clique, clique_number, maximum_independent_set, independence_number, all_subsets_checked, certificate).

View answer and explanation([0, 1, 2], 3, [0, 3, 5], 3, Yes, exhaustive_all_vertex_subsets)

Inspect all vertex subsets. Return a maximum clique and independent set, preserving the first maximum witness in increasing vertex-bitmask order, not re-sorting tied witnesses lexicographically. This bounded exhaustive certificate is not a large-graph efficiency claim. The exact result, retaining the requested order, is ([0, 1, 2], 3, [0, 3, 5], 3, Yes, exhaustive_all_vertex_subsets).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0055/)

Cliques and independent sets · qf-mod37-clique-exact-extrema · candidate





## 56. Exact Extremal Sizes Need the Complete Vertex-Subset Search

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [1, 2], [2, 4], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (maximum_clique, clique_number, maximum_independent_set, independence_number, all_subsets_checked, certificate).

View answer and explanation([0, 1, 2], 3, [1, 3, 5], 3, Yes, exhaustive_all_vertex_subsets)

Inspect all vertex subsets. Return a maximum clique and independent set, preserving the first maximum witness in increasing vertex-bitmask order, not re-sorting tied witnesses lexicographically. This bounded exhaustive certificate is not a large-graph efficiency claim. The exact result, retaining the requested order, is ([0, 1, 2], 3, [1, 3, 5], 3, Yes, exhaustive_all_vertex_subsets).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0056/)

Cliques and independent sets · qf-mod37-clique-exact-extrema · candidate





## 57. Exact Extremal Sizes Need the Complete Vertex-Subset Search

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [1, 2], [1, 3], [2, 4], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (maximum_clique, clique_number, maximum_independent_set, independence_number, all_subsets_checked, certificate).

View answer and explanation([0, 1, 2], 3, [0, 3, 5], 3, Yes, exhaustive_all_vertex_subsets)

Inspect all vertex subsets. Return a maximum clique and independent set, preserving the first maximum witness in increasing vertex-bitmask order, not re-sorting tied witnesses lexicographically. This bounded exhaustive certificate is not a large-graph efficiency claim. The exact result, retaining the requested order, is ([0, 1, 2], 3, [0, 3, 5], 3, Yes, exhaustive_all_vertex_subsets).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0057/)

Cliques and independent sets · qf-mod37-clique-exact-extrema · candidate





## 58. Exact Extremal Sizes Need the Complete Vertex-Subset Search

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [2, 4], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (maximum_clique, clique_number, maximum_independent_set, independence_number, all_subsets_checked, certificate).

View answer and explanation([0, 1, 2], 3, [2, 3, 5], 3, Yes, exhaustive_all_vertex_subsets)

Inspect all vertex subsets. Return a maximum clique and independent set, preserving the first maximum witness in increasing vertex-bitmask order, not re-sorting tied witnesses lexicographically. This bounded exhaustive certificate is not a large-graph efficiency claim. The exact result, retaining the requested order, is ([0, 1, 2], 3, [2, 3, 5], 3, Yes, exhaustive_all_vertex_subsets).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0058/)

Cliques and independent sets · qf-mod37-clique-exact-extrema · candidate





## 59. Exact Extremal Sizes Need the Complete Vertex-Subset Search

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 5], [1, 2], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (maximum_clique, clique_number, maximum_independent_set, independence_number, all_subsets_checked, certificate).

View answer and explanation([0, 1, 2], 3, [1, 3, 5], 3, Yes, exhaustive_all_vertex_subsets)

Inspect all vertex subsets. Return a maximum clique and independent set, preserving the first maximum witness in increasing vertex-bitmask order, not re-sorting tied witnesses lexicographically. This bounded exhaustive certificate is not a large-graph efficiency claim. The exact result, retaining the requested order, is ([0, 1, 2], 3, [1, 3, 5], 3, Yes, exhaustive_all_vertex_subsets).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0059/)

Cliques and independent sets · qf-mod37-clique-exact-extrema · candidate





## 60. Exact Extremal Sizes Need the Complete Vertex-Subset Search

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [0, 5], [1, 2], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (maximum_clique, clique_number, maximum_independent_set, independence_number, all_subsets_checked, certificate).

View answer and explanation([0, 1, 2], 3, [1, 3, 5], 3, Yes, exhaustive_all_vertex_subsets)

Inspect all vertex subsets. Return a maximum clique and independent set, preserving the first maximum witness in increasing vertex-bitmask order, not re-sorting tied witnesses lexicographically. This bounded exhaustive certificate is not a large-graph efficiency claim. The exact result, retaining the requested order, is ([0, 1, 2], 3, [1, 3, 5], 3, Yes, exhaustive_all_vertex_subsets).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0060/)

Cliques and independent sets · qf-mod37-clique-exact-extrema · candidate





## 61. Exact Extremal Sizes Need the Complete Vertex-Subset Search

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 5], [1, 2], [1, 3], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (maximum_clique, clique_number, maximum_independent_set, independence_number, all_subsets_checked, certificate).

View answer and explanation([0, 1, 2], 3, [2, 3, 5], 3, Yes, exhaustive_all_vertex_subsets)

Inspect all vertex subsets. Return a maximum clique and independent set, preserving the first maximum witness in increasing vertex-bitmask order, not re-sorting tied witnesses lexicographically. This bounded exhaustive certificate is not a large-graph efficiency claim. The exact result, retaining the requested order, is ([0, 1, 2], 3, [2, 3, 5], 3, Yes, exhaustive_all_vertex_subsets).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0061/)

Cliques and independent sets · qf-mod37-clique-exact-extrema · candidate





## 62. Exact Extremal Sizes Need the Complete Vertex-Subset Search

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [0, 5], [1, 2], [1, 3], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (maximum_clique, clique_number, maximum_independent_set, independence_number, all_subsets_checked, certificate).

View answer and explanation([0, 1, 2], 3, [2, 3, 5], 3, Yes, exhaustive_all_vertex_subsets)

Inspect all vertex subsets. Return a maximum clique and independent set, preserving the first maximum witness in increasing vertex-bitmask order, not re-sorting tied witnesses lexicographically. This bounded exhaustive certificate is not a large-graph efficiency claim. The exact result, retaining the requested order, is ([0, 1, 2], 3, [2, 3, 5], 3, Yes, exhaustive_all_vertex_subsets).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0062/)

Cliques and independent sets · qf-mod37-clique-exact-extrema · candidate





## 63. Exact Extremal Sizes Need the Complete Vertex-Subset Search

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 5], [1, 2], [2, 4], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (maximum_clique, clique_number, maximum_independent_set, independence_number, all_subsets_checked, certificate).

View answer and explanation([0, 1, 2], 3, [1, 3, 5], 3, Yes, exhaustive_all_vertex_subsets)

Inspect all vertex subsets. Return a maximum clique and independent set, preserving the first maximum witness in increasing vertex-bitmask order, not re-sorting tied witnesses lexicographically. This bounded exhaustive certificate is not a large-graph efficiency claim. The exact result, retaining the requested order, is ([0, 1, 2], 3, [1, 3, 5], 3, Yes, exhaustive_all_vertex_subsets).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0063/)

Cliques and independent sets · qf-mod37-clique-exact-extrema · candidate





## 64. Exact Extremal Sizes Need the Complete Vertex-Subset Search

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [0, 5], [1, 2], [2, 4], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (maximum_clique, clique_number, maximum_independent_set, independence_number, all_subsets_checked, certificate).

View answer and explanation([0, 1, 2], 3, [1, 3, 5], 3, Yes, exhaustive_all_vertex_subsets)

Inspect all vertex subsets. Return a maximum clique and independent set, preserving the first maximum witness in increasing vertex-bitmask order, not re-sorting tied witnesses lexicographically. This bounded exhaustive certificate is not a large-graph efficiency claim. The exact result, retaining the requested order, is ([0, 1, 2], 3, [1, 3, 5], 3, Yes, exhaustive_all_vertex_subsets).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0064/)

Cliques and independent sets · qf-mod37-clique-exact-extrema · candidate





## 65. Exact Extremal Sizes Need the Complete Vertex-Subset Search

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 5], [1, 2], [1, 3], [2, 4], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (maximum_clique, clique_number, maximum_independent_set, independence_number, all_subsets_checked, certificate).

View answer and explanation([0, 1, 2], 3, [2, 3, 5], 3, Yes, exhaustive_all_vertex_subsets)

Inspect all vertex subsets. Return a maximum clique and independent set, preserving the first maximum witness in increasing vertex-bitmask order, not re-sorting tied witnesses lexicographically. This bounded exhaustive certificate is not a large-graph efficiency claim. The exact result, retaining the requested order, is ([0, 1, 2], 3, [2, 3, 5], 3, Yes, exhaustive_all_vertex_subsets).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0065/)

Cliques and independent sets · qf-mod37-clique-exact-extrema · candidate





## 66. Exact Extremal Sizes Need the Complete Vertex-Subset Search

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [0, 5], [1, 2], [1, 3], [2, 4], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (maximum_clique, clique_number, maximum_independent_set, independence_number, all_subsets_checked, certificate).

View answer and explanation([0, 1, 2], 3, [2, 3, 5], 3, Yes, exhaustive_all_vertex_subsets)

Inspect all vertex subsets. Return a maximum clique and independent set, preserving the first maximum witness in increasing vertex-bitmask order, not re-sorting tied witnesses lexicographically. This bounded exhaustive certificate is not a large-graph efficiency claim. The exact result, retaining the requested order, is ([0, 1, 2], 3, [2, 3, 5], 3, Yes, exhaustive_all_vertex_subsets).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0066/)

Cliques and independent sets · qf-mod37-clique-exact-extrema · candidate





## 67. Exact Extremal Sizes Need the Complete Vertex-Subset Search

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [1, 2], [1, 5], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (maximum_clique, clique_number, maximum_independent_set, independence_number, all_subsets_checked, certificate).

View answer and explanation([0, 1, 2], 3, [0, 3, 5], 3, Yes, exhaustive_all_vertex_subsets)

Inspect all vertex subsets. Return a maximum clique and independent set, preserving the first maximum witness in increasing vertex-bitmask order, not re-sorting tied witnesses lexicographically. This bounded exhaustive certificate is not a large-graph efficiency claim. The exact result, retaining the requested order, is ([0, 1, 2], 3, [0, 3, 5], 3, Yes, exhaustive_all_vertex_subsets).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0067/)

Cliques and independent sets · qf-mod37-clique-exact-extrema · candidate





## 68. Exact Extremal Sizes Need the Complete Vertex-Subset Search

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 5], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (maximum_clique, clique_number, maximum_independent_set, independence_number, all_subsets_checked, certificate).

View answer and explanation([0, 1, 2], 3, [2, 3, 5], 3, Yes, exhaustive_all_vertex_subsets)

Inspect all vertex subsets. Return a maximum clique and independent set, preserving the first maximum witness in increasing vertex-bitmask order, not re-sorting tied witnesses lexicographically. This bounded exhaustive certificate is not a large-graph efficiency claim. The exact result, retaining the requested order, is ([0, 1, 2], 3, [2, 3, 5], 3, Yes, exhaustive_all_vertex_subsets).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0068/)

Cliques and independent sets · qf-mod37-clique-exact-extrema · candidate





## 69. Exact Extremal Sizes Need the Complete Vertex-Subset Search

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [1, 2], [1, 3], [1, 5], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (maximum_clique, clique_number, maximum_independent_set, independence_number, all_subsets_checked, certificate).

View answer and explanation([0, 1, 2], 3, [0, 3, 5], 3, Yes, exhaustive_all_vertex_subsets)

Inspect all vertex subsets. Return a maximum clique and independent set, preserving the first maximum witness in increasing vertex-bitmask order, not re-sorting tied witnesses lexicographically. This bounded exhaustive certificate is not a large-graph efficiency claim. The exact result, retaining the requested order, is ([0, 1, 2], 3, [0, 3, 5], 3, Yes, exhaustive_all_vertex_subsets).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0069/)

Cliques and independent sets · qf-mod37-clique-exact-extrema · candidate





## 70. Exact Extremal Sizes Need the Complete Vertex-Subset Search

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [1, 5], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (maximum_clique, clique_number, maximum_independent_set, independence_number, all_subsets_checked, certificate).

View answer and explanation([0, 1, 2], 3, [2, 3, 5], 3, Yes, exhaustive_all_vertex_subsets)

Inspect all vertex subsets. Return a maximum clique and independent set, preserving the first maximum witness in increasing vertex-bitmask order, not re-sorting tied witnesses lexicographically. This bounded exhaustive certificate is not a large-graph efficiency claim. The exact result, retaining the requested order, is ([0, 1, 2], 3, [2, 3, 5], 3, Yes, exhaustive_all_vertex_subsets).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0070/)

Cliques and independent sets · qf-mod37-clique-exact-extrema · candidate





## 71. Exact Extremal Sizes Need the Complete Vertex-Subset Search

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [1, 2], [1, 5], [2, 4], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (maximum_clique, clique_number, maximum_independent_set, independence_number, all_subsets_checked, certificate).

View answer and explanation([0, 1, 2], 3, [0, 3, 5], 3, Yes, exhaustive_all_vertex_subsets)

Inspect all vertex subsets. Return a maximum clique and independent set, preserving the first maximum witness in increasing vertex-bitmask order, not re-sorting tied witnesses lexicographically. This bounded exhaustive certificate is not a large-graph efficiency claim. The exact result, retaining the requested order, is ([0, 1, 2], 3, [0, 3, 5], 3, Yes, exhaustive_all_vertex_subsets).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0071/)

Cliques and independent sets · qf-mod37-clique-exact-extrema · candidate





## 72. Exact Extremal Sizes Need the Complete Vertex-Subset Search

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 5], [2, 4], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (maximum_clique, clique_number, maximum_independent_set, independence_number, all_subsets_checked, certificate).

View answer and explanation([0, 1, 2], 3, [2, 3, 5], 3, Yes, exhaustive_all_vertex_subsets)

Inspect all vertex subsets. Return a maximum clique and independent set, preserving the first maximum witness in increasing vertex-bitmask order, not re-sorting tied witnesses lexicographically. This bounded exhaustive certificate is not a large-graph efficiency claim. The exact result, retaining the requested order, is ([0, 1, 2], 3, [2, 3, 5], 3, Yes, exhaustive_all_vertex_subsets).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0072/)

Cliques and independent sets · qf-mod37-clique-exact-extrema · candidate





## 73. Exact Extremal Sizes Need the Complete Vertex-Subset Search

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [1, 2], [1, 3], [1, 5], [2, 4], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (maximum_clique, clique_number, maximum_independent_set, independence_number, all_subsets_checked, certificate).

View answer and explanation([0, 1, 2], 3, [0, 3, 5], 3, Yes, exhaustive_all_vertex_subsets)

Inspect all vertex subsets. Return a maximum clique and independent set, preserving the first maximum witness in increasing vertex-bitmask order, not re-sorting tied witnesses lexicographically. This bounded exhaustive certificate is not a large-graph efficiency claim. The exact result, retaining the requested order, is ([0, 1, 2], 3, [0, 3, 5], 3, Yes, exhaustive_all_vertex_subsets).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0073/)

Cliques and independent sets · qf-mod37-clique-exact-extrema · candidate





## 74. Exact Extremal Sizes Need the Complete Vertex-Subset Search

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [1, 5], [2, 4], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (maximum_clique, clique_number, maximum_independent_set, independence_number, all_subsets_checked, certificate).

View answer and explanation([0, 1, 2], 3, [2, 3, 5], 3, Yes, exhaustive_all_vertex_subsets)

Inspect all vertex subsets. Return a maximum clique and independent set, preserving the first maximum witness in increasing vertex-bitmask order, not re-sorting tied witnesses lexicographically. This bounded exhaustive certificate is not a large-graph efficiency claim. The exact result, retaining the requested order, is ([0, 1, 2], 3, [2, 3, 5], 3, Yes, exhaustive_all_vertex_subsets).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0074/)

Cliques and independent sets · qf-mod37-clique-exact-extrema · candidate





## 75. Exact Extremal Sizes Need the Complete Vertex-Subset Search

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 5], [1, 2], [1, 5], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (maximum_clique, clique_number, maximum_independent_set, independence_number, all_subsets_checked, certificate).

View answer and explanation([0, 1, 2], 3, [2, 3, 5], 3, Yes, exhaustive_all_vertex_subsets)

Inspect all vertex subsets. Return a maximum clique and independent set, preserving the first maximum witness in increasing vertex-bitmask order, not re-sorting tied witnesses lexicographically. This bounded exhaustive certificate is not a large-graph efficiency claim. The exact result, retaining the requested order, is ([0, 1, 2], 3, [2, 3, 5], 3, Yes, exhaustive_all_vertex_subsets).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0075/)

Cliques and independent sets · qf-mod37-clique-exact-extrema · candidate





## 76. Validity, Inclusion-Maximality and Global Maximum Are Separate

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [1, 2], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 4], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Requested kind=clique. Give (kind, vertices, valid, maximal, maximum_size, is_maximum, boundary).

View answer and explanation(clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)

A valid candidate is maximal if no outside vertex can be added while preserving the property. It is maximum only if valid and its size equals the independently computed global maximum. An invalid candidate cannot become maximum by matching the maximum size. A clique gives only a chromatic lower bound in general. The exact result, retaining the requested order, is (clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0076/)

Cliques and independent sets · qf-mod37-clique-maximal-vs-maximum · candidate





## 77. Validity, Inclusion-Maximality and Global Maximum Are Separate

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [1, 2], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 4], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Requested kind=clique. Give (kind, vertices, valid, maximal, maximum_size, is_maximum, boundary).

View answer and explanation(clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)

A valid candidate is maximal if no outside vertex can be added while preserving the property. It is maximum only if valid and its size equals the independently computed global maximum. An invalid candidate cannot become maximum by matching the maximum size. A clique gives only a chromatic lower bound in general. The exact result, retaining the requested order, is (clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0077/)

Cliques and independent sets · qf-mod37-clique-maximal-vs-maximum · candidate





## 78. Validity, Inclusion-Maximality and Global Maximum Are Separate

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [1, 2], [1, 3], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 4], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Requested kind=clique. Give (kind, vertices, valid, maximal, maximum_size, is_maximum, boundary).

View answer and explanation(clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)

A valid candidate is maximal if no outside vertex can be added while preserving the property. It is maximum only if valid and its size equals the independently computed global maximum. An invalid candidate cannot become maximum by matching the maximum size. A clique gives only a chromatic lower bound in general. The exact result, retaining the requested order, is (clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0078/)

Cliques and independent sets · qf-mod37-clique-maximal-vs-maximum · candidate





## 79. Validity, Inclusion-Maximality and Global Maximum Are Separate

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 4], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Requested kind=clique. Give (kind, vertices, valid, maximal, maximum_size, is_maximum, boundary).

View answer and explanation(clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)

A valid candidate is maximal if no outside vertex can be added while preserving the property. It is maximum only if valid and its size equals the independently computed global maximum. An invalid candidate cannot become maximum by matching the maximum size. A clique gives only a chromatic lower bound in general. The exact result, retaining the requested order, is (clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0079/)

Cliques and independent sets · qf-mod37-clique-maximal-vs-maximum · candidate





## 80. Validity, Inclusion-Maximality and Global Maximum Are Separate

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [1, 2], [2, 4], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 4], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Requested kind=clique. Give (kind, vertices, valid, maximal, maximum_size, is_maximum, boundary).

View answer and explanation(clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)

A valid candidate is maximal if no outside vertex can be added while preserving the property. It is maximum only if valid and its size equals the independently computed global maximum. An invalid candidate cannot become maximum by matching the maximum size. A clique gives only a chromatic lower bound in general. The exact result, retaining the requested order, is (clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0080/)

Cliques and independent sets · qf-mod37-clique-maximal-vs-maximum · candidate





## 81. Validity, Inclusion-Maximality and Global Maximum Are Separate

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [1, 2], [2, 4], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 4], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Requested kind=clique. Give (kind, vertices, valid, maximal, maximum_size, is_maximum, boundary).

View answer and explanation(clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)

A valid candidate is maximal if no outside vertex can be added while preserving the property. It is maximum only if valid and its size equals the independently computed global maximum. An invalid candidate cannot become maximum by matching the maximum size. A clique gives only a chromatic lower bound in general. The exact result, retaining the requested order, is (clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0081/)

Cliques and independent sets · qf-mod37-clique-maximal-vs-maximum · candidate





## 82. Validity, Inclusion-Maximality and Global Maximum Are Separate

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [1, 2], [1, 3], [2, 4], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 4], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Requested kind=clique. Give (kind, vertices, valid, maximal, maximum_size, is_maximum, boundary).

View answer and explanation(clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)

A valid candidate is maximal if no outside vertex can be added while preserving the property. It is maximum only if valid and its size equals the independently computed global maximum. An invalid candidate cannot become maximum by matching the maximum size. A clique gives only a chromatic lower bound in general. The exact result, retaining the requested order, is (clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0082/)

Cliques and independent sets · qf-mod37-clique-maximal-vs-maximum · candidate





## 83. Validity, Inclusion-Maximality and Global Maximum Are Separate

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [2, 4], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 4], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Requested kind=clique. Give (kind, vertices, valid, maximal, maximum_size, is_maximum, boundary).

View answer and explanation(clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)

A valid candidate is maximal if no outside vertex can be added while preserving the property. It is maximum only if valid and its size equals the independently computed global maximum. An invalid candidate cannot become maximum by matching the maximum size. A clique gives only a chromatic lower bound in general. The exact result, retaining the requested order, is (clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0083/)

Cliques and independent sets · qf-mod37-clique-maximal-vs-maximum · candidate





## 84. Validity, Inclusion-Maximality and Global Maximum Are Separate

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 5], [1, 2], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 4], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Requested kind=clique. Give (kind, vertices, valid, maximal, maximum_size, is_maximum, boundary).

View answer and explanation(clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)

A valid candidate is maximal if no outside vertex can be added while preserving the property. It is maximum only if valid and its size equals the independently computed global maximum. An invalid candidate cannot become maximum by matching the maximum size. A clique gives only a chromatic lower bound in general. The exact result, retaining the requested order, is (clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0084/)

Cliques and independent sets · qf-mod37-clique-maximal-vs-maximum · candidate





## 85. Validity, Inclusion-Maximality and Global Maximum Are Separate

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [0, 5], [1, 2], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 4], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Requested kind=clique. Give (kind, vertices, valid, maximal, maximum_size, is_maximum, boundary).

View answer and explanation(clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)

A valid candidate is maximal if no outside vertex can be added while preserving the property. It is maximum only if valid and its size equals the independently computed global maximum. An invalid candidate cannot become maximum by matching the maximum size. A clique gives only a chromatic lower bound in general. The exact result, retaining the requested order, is (clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0085/)

Cliques and independent sets · qf-mod37-clique-maximal-vs-maximum · candidate





## 86. Validity, Inclusion-Maximality and Global Maximum Are Separate

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 5], [1, 2], [1, 3], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 4], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Requested kind=clique. Give (kind, vertices, valid, maximal, maximum_size, is_maximum, boundary).

View answer and explanation(clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)

A valid candidate is maximal if no outside vertex can be added while preserving the property. It is maximum only if valid and its size equals the independently computed global maximum. An invalid candidate cannot become maximum by matching the maximum size. A clique gives only a chromatic lower bound in general. The exact result, retaining the requested order, is (clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0086/)

Cliques and independent sets · qf-mod37-clique-maximal-vs-maximum · candidate





## 87. Validity, Inclusion-Maximality and Global Maximum Are Separate

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [0, 5], [1, 2], [1, 3], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 4], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Requested kind=clique. Give (kind, vertices, valid, maximal, maximum_size, is_maximum, boundary).

View answer and explanation(clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)

A valid candidate is maximal if no outside vertex can be added while preserving the property. It is maximum only if valid and its size equals the independently computed global maximum. An invalid candidate cannot become maximum by matching the maximum size. A clique gives only a chromatic lower bound in general. The exact result, retaining the requested order, is (clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0087/)

Cliques and independent sets · qf-mod37-clique-maximal-vs-maximum · candidate





## 88. Validity, Inclusion-Maximality and Global Maximum Are Separate

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 5], [1, 2], [2, 4], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 4], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Requested kind=clique. Give (kind, vertices, valid, maximal, maximum_size, is_maximum, boundary).

View answer and explanation(clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)

A valid candidate is maximal if no outside vertex can be added while preserving the property. It is maximum only if valid and its size equals the independently computed global maximum. An invalid candidate cannot become maximum by matching the maximum size. A clique gives only a chromatic lower bound in general. The exact result, retaining the requested order, is (clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0088/)

Cliques and independent sets · qf-mod37-clique-maximal-vs-maximum · candidate





## 89. Validity, Inclusion-Maximality and Global Maximum Are Separate

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [0, 5], [1, 2], [2, 4], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 4], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Requested kind=clique. Give (kind, vertices, valid, maximal, maximum_size, is_maximum, boundary).

View answer and explanation(clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)

A valid candidate is maximal if no outside vertex can be added while preserving the property. It is maximum only if valid and its size equals the independently computed global maximum. An invalid candidate cannot become maximum by matching the maximum size. A clique gives only a chromatic lower bound in general. The exact result, retaining the requested order, is (clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0089/)

Cliques and independent sets · qf-mod37-clique-maximal-vs-maximum · candidate





## 90. Validity, Inclusion-Maximality and Global Maximum Are Separate

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 5], [1, 2], [1, 3], [2, 4], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 4], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Requested kind=clique. Give (kind, vertices, valid, maximal, maximum_size, is_maximum, boundary).

View answer and explanation(clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)

A valid candidate is maximal if no outside vertex can be added while preserving the property. It is maximum only if valid and its size equals the independently computed global maximum. An invalid candidate cannot become maximum by matching the maximum size. A clique gives only a chromatic lower bound in general. The exact result, retaining the requested order, is (clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0090/)

Cliques and independent sets · qf-mod37-clique-maximal-vs-maximum · candidate





## 91. Validity, Inclusion-Maximality and Global Maximum Are Separate

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [0, 5], [1, 2], [1, 3], [2, 4], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 4], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Requested kind=clique. Give (kind, vertices, valid, maximal, maximum_size, is_maximum, boundary).

View answer and explanation(clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)

A valid candidate is maximal if no outside vertex can be added while preserving the property. It is maximum only if valid and its size equals the independently computed global maximum. An invalid candidate cannot become maximum by matching the maximum size. A clique gives only a chromatic lower bound in general. The exact result, retaining the requested order, is (clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0091/)

Cliques and independent sets · qf-mod37-clique-maximal-vs-maximum · candidate





## 92. Validity, Inclusion-Maximality and Global Maximum Are Separate

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [1, 2], [1, 5], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 4], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Requested kind=clique. Give (kind, vertices, valid, maximal, maximum_size, is_maximum, boundary).

View answer and explanation(clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)

A valid candidate is maximal if no outside vertex can be added while preserving the property. It is maximum only if valid and its size equals the independently computed global maximum. An invalid candidate cannot become maximum by matching the maximum size. A clique gives only a chromatic lower bound in general. The exact result, retaining the requested order, is (clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0092/)

Cliques and independent sets · qf-mod37-clique-maximal-vs-maximum · candidate





## 93. Validity, Inclusion-Maximality and Global Maximum Are Separate

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 5], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 4], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Requested kind=clique. Give (kind, vertices, valid, maximal, maximum_size, is_maximum, boundary).

View answer and explanation(clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)

A valid candidate is maximal if no outside vertex can be added while preserving the property. It is maximum only if valid and its size equals the independently computed global maximum. An invalid candidate cannot become maximum by matching the maximum size. A clique gives only a chromatic lower bound in general. The exact result, retaining the requested order, is (clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0093/)

Cliques and independent sets · qf-mod37-clique-maximal-vs-maximum · candidate





## 94. Validity, Inclusion-Maximality and Global Maximum Are Separate

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [1, 2], [1, 3], [1, 5], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 4], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Requested kind=clique. Give (kind, vertices, valid, maximal, maximum_size, is_maximum, boundary).

View answer and explanation(clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)

A valid candidate is maximal if no outside vertex can be added while preserving the property. It is maximum only if valid and its size equals the independently computed global maximum. An invalid candidate cannot become maximum by matching the maximum size. A clique gives only a chromatic lower bound in general. The exact result, retaining the requested order, is (clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0094/)

Cliques and independent sets · qf-mod37-clique-maximal-vs-maximum · candidate





## 95. Validity, Inclusion-Maximality and Global Maximum Are Separate

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [1, 5], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 4], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Requested kind=clique. Give (kind, vertices, valid, maximal, maximum_size, is_maximum, boundary).

View answer and explanation(clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)

A valid candidate is maximal if no outside vertex can be added while preserving the property. It is maximum only if valid and its size equals the independently computed global maximum. An invalid candidate cannot become maximum by matching the maximum size. A clique gives only a chromatic lower bound in general. The exact result, retaining the requested order, is (clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0095/)

Cliques and independent sets · qf-mod37-clique-maximal-vs-maximum · candidate





## 96. Validity, Inclusion-Maximality and Global Maximum Are Separate

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [1, 2], [1, 5], [2, 4], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 4], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Requested kind=clique. Give (kind, vertices, valid, maximal, maximum_size, is_maximum, boundary).

View answer and explanation(clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)

A valid candidate is maximal if no outside vertex can be added while preserving the property. It is maximum only if valid and its size equals the independently computed global maximum. An invalid candidate cannot become maximum by matching the maximum size. A clique gives only a chromatic lower bound in general. The exact result, retaining the requested order, is (clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0096/)

Cliques and independent sets · qf-mod37-clique-maximal-vs-maximum · candidate





## 97. Validity, Inclusion-Maximality and Global Maximum Are Separate

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 5], [2, 4], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 4], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Requested kind=clique. Give (kind, vertices, valid, maximal, maximum_size, is_maximum, boundary).

View answer and explanation(clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)

A valid candidate is maximal if no outside vertex can be added while preserving the property. It is maximum only if valid and its size equals the independently computed global maximum. An invalid candidate cannot become maximum by matching the maximum size. A clique gives only a chromatic lower bound in general. The exact result, retaining the requested order, is (clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0097/)

Cliques and independent sets · qf-mod37-clique-maximal-vs-maximum · candidate





## 98. Validity, Inclusion-Maximality and Global Maximum Are Separate

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [1, 2], [1, 3], [1, 5], [2, 4], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 4], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Requested kind=clique. Give (kind, vertices, valid, maximal, maximum_size, is_maximum, boundary).

View answer and explanation(clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)

A valid candidate is maximal if no outside vertex can be added while preserving the property. It is maximum only if valid and its size equals the independently computed global maximum. An invalid candidate cannot become maximum by matching the maximum size. A clique gives only a chromatic lower bound in general. The exact result, retaining the requested order, is (clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0098/)

Cliques and independent sets · qf-mod37-clique-maximal-vs-maximum · candidate





## 99. Validity, Inclusion-Maximality and Global Maximum Are Separate

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [1, 5], [2, 4], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 4], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Requested kind=clique. Give (kind, vertices, valid, maximal, maximum_size, is_maximum, boundary).

View answer and explanation(clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)

A valid candidate is maximal if no outside vertex can be added while preserving the property. It is maximum only if valid and its size equals the independently computed global maximum. An invalid candidate cannot become maximum by matching the maximum size. A clique gives only a chromatic lower bound in general. The exact result, retaining the requested order, is (clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0099/)

Cliques and independent sets · qf-mod37-clique-maximal-vs-maximum · candidate





## 100. Validity, Inclusion-Maximality and Global Maximum Are Separate

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 5], [1, 2], [1, 5], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 4], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Requested kind=clique. Give (kind, vertices, valid, maximal, maximum_size, is_maximum, boundary).

View answer and explanation(clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)

A valid candidate is maximal if no outside vertex can be added while preserving the property. It is maximum only if valid and its size equals the independently computed global maximum. An invalid candidate cannot become maximum by matching the maximum size. A clique gives only a chromatic lower bound in general. The exact result, retaining the requested order, is (clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r02-clique-and-independent-set-0100/)

Cliques and independent sets · qf-mod37-clique-maximal-vs-maximum · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations/validation-report.json)