Exponential upper-bound constant for diagonal Ramsey numbers
What is the least constant c such that R(k,k) ≤ c^(k+o(k))?
Model / AI: GPT-5.4 Pro, GPT-5.6 Sol Max, Claude
•self-reported!disputed (certificate)
2026
HorizonMath certificate for c ≤ 3.6961
Model / AI: GPT-5.4 Pro, GPT-5.6 Sol Max
•self-reported
HorizonMath (arXiv 2603.15617v2, Appendix A.3) credits GPT-5.4 Pro, reproduced by GPT-5.6 Sol Max, with a certificate in the framework of Gupta, Ndiaye, Norin and Wei (their Theorem 14) giving R(k,k) ≤ 3.6960839126^(k+o(k)), below that paper's 3.7992. The benchmark's validator, validators/ramsey_asymptotic.py, accepts the certificate.
Source review pending. Source review pending. Submitted by the author of the audit, Carlos Toledo (carlostoledo1891), in issue #2.
An independent exact audit (cert-machine, Carlos Toledo) found that at λ = 1, where the constant is read, the certificate uses (X(1), Y(1)) = (0.22745, 0.9988). Theorem 14 needs that pair in the region R, and it lies outside: Erdős's 1947 random-colouring bound at e = 13/2000, p = 11981/500000 exceeds the pair's rate by 0.00130. Of the 201 decided points, 56 lie outside R and 128 more are not placed in R by the bound the problem names. The validator (line 299, min) accepts a pair when either of two orientation inequalities holds, while membership in R needs both; with max instead, it rejects the certificate on its first interval. The audit tooling was built with AI assistance. The finding was reported to the authors as HorizonMath issue #10 on 2026-10-05.
Source review pending. Source review pending. Submitted by the author of the audit, Carlos Toledo (carlostoledo1891), in issue #2; the authors had not replied to issue #10 at submission.