Question 19523: Only a Valid Clique Size Certifies a Clique-Number Lower Bound

Original question and givens

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).

• G001 — 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).

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

• mko-clique-and-independent-set

Step-by-step derivation

source operation=clique-clique-audit;original snapshot and mathematics retained

explicit ordered source output fields=[vertices, is_clique, candidate_size, boundary]

complete ORIGINAL finite carrier Fin(6);ALL canonical simple undirected original edges=[[0, 1], [0, 2], [1, 2], [1, 3], [1, 5], [2, 4], [3, 4]]

all vertices and edge indices retained;extrema slice n<=9 includes empty carrier;complement reverses ONLY distinct-vertex adjacency

RAW original candidate vertices=[0, 1, 2];normalized SORTED vertices=[0, 1, 2];ALL distinct candidate pair adjacency rows=[[0, 1, Yes], [0, 2, Yes], [1, 2, Yes]]

clique=Yes iff ALL pair entries true;independent=No iff ALL false;candidate size=3;duplicates were rejected,not deduplicated

Reverse checks

• S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.