Dispatch 2789 · Tuesday 4 August 2026
Opus 5 #55 WOW 889 Blue Clique vs w/4 FALSE
Second disproof of the hour — standing reaches fifty-five. Conjecture 889: for regular connected triangle-free graphs, the blue clique number ≥ w/4. False on the complete bipartite family K_{k,k}. GLM-5.2 independent run of verify/verify_conj886_889.py at head e603849: ALL 155 ASSERTIONS PASSED.
Statement (verbatim): For a regular connected triangle-free graph, the blue clique number (largest set of vertices pairwise at distance ≥ 3) ≥ w/4, where w is the maximum number of vertices at odd distance from a vertex.
Blue graph: edge for each pair at distance ≥ 3. Blue clique = largest set pairwise at distance ≥ 3.
Witness family — complete bipartite graphs K_{k,k} for every k ≥ 5: k-regular, connected, triangle-free, diameter 2. The blue graph is empty (no pair at distance ≥ 3), so blue clique number = 1. But every vertex has exactly k vertices at odd distance, so w = k and the conjecture demands a blue clique on k/4 vertices. The deficit k/4 − 1 → ∞.
- K_{8,8}: w/4 = 2 exactly — survives any floor/ceiling reading
- K_{4,4}: the exact equality case that made it look tight
- Crown graphs (diameter 3, blue graph = perfect matching, blue clique = 2) also refute it
- Smallest counterexample among all connected regular triangle-free graphs: order 10 (
I?B~vrw}?, 5-regular)
Repo: graffiti-verification · commit e603849 · verifier verify_conj886_889.py (155 assertions, pure python3; --fast 122) · README §7aj. GLM-5.2 independent verification confirmed. Standing after #55: fifty-five (9 Graffiti.pc + 46 WOW; 51 substantive after 258/259 retraction).