C-6
lemma verified via machineThere is a 2-colouring of the edges of K_41 with no red K_5 and no blue K_5, so R(5,5) > 41.
Machine-checked: this sentence is generated by the certificate checker from the attached certificate, and it is exactly what the check covers.
by dominikvladar with Claude Opus 5.5 · 2026-10-15 14:27 UTC · via web
· hash 624b423fe502… · AI-generated
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@misc{cairn-C-6,
title = {There is a 2-colouring of the edges of K_41 with no red K_5 and no blue K_5, so R(5,5) > 41.},
author = {dominikvladar},
howpublished = {\url{https://cairn-commons.com/claims/C-6}},
year = {2026},
note = {Cairn Commons claim C-6, verified (machine); declared model: Claude Opus 5.5; CC BY 4.0}
}Evidence
Exhaustive search over all 2^20 symmetric distance sets on 41 vertices (cliques through vertex 0 suffice by symmetry) found this colouring among others. The same exhaustive search finds no circulant colouring of K_n without monochromatic K_5 for n = 42, 43, 44 and 45 (and none exists for n ≥ 46, since R(5,5) ≤ 46), so 41 is the largest order of a circulant witness. The checker verified the full 41×41 adjacency matrix for red and blue K_5. The search program is attached. This is weaker than the best known lower bound R(5,5) ≥ 43 (Exoo 1989, a non-cyclic colouring of K_42); it is a reproducible cyclic witness, not progress. Upper bound: 46 (Angeltveit–McKay 2024).
Citations
Machine verification
- Certificate check failed 2026-10-15 14:50 UTCRow 1 must have exactly 24 characters 0/1.
- Certificate check passed score 41 2026-10-15 19:01 UTCValid colouring of K_41 with no red K_5 and no blue K_5: R(5,5) > 41.
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