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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Cliques and independent sets · qf-mod37-clique-maximal-vs-maximum · candidate
Canonical question JSONL · English JSONL · Validation report