Resolves YES if, by December 31 2026, the proof that R(B_{n-1}, B_n) = 4n-1 for all n where q = 4n-1 is prime with q ≡ 3 (mod 8) is confirmed as correct by a qualified reviewer (e.g., published, accepted by a journal, or endorsed as correct by a Ramsey theory researcher such as the problem contributor William J. Wesley). Resolves NO if a fatal gap is identified that cannot be repaired. Resolves N/A if no qualified review occurs by the resolution date, or if reviewer feedback is ambiguous (e.g., "probably correct but I haven't checked every step").
What this is: A proof draft claiming a new infinite family of exact book Ramsey numbers. The construction is a 2-block circulant graph built from quadratic residues mod a prime q = 4n-1, and the proof verifies six codegree bounds to show the graph avoids B_{n-1} while its complement avoids B_n. This extends prior work by Wesley (2024) and Lidický–McKinley–Pfender–Van Overberghe (2024), who established the result when 2n-1 is a prime power ≡ 1 (mod 4) and for all n ≤ 21. The new family includes cases not covered by prior work, such as n = 35, 53, 71, 77, 83, 95.
Why I think it may be correct:
The proof was independently reviewed by GPT-5.4 Pro acting as a referee, which identified a missing lemma in an earlier draft. Claude Opus 4.6 independently produced the same fix. The current draft incorporates both.
The algebraic construction has been computationally verified for all eligible primes through n = 143, and all instances satisfy the required codegree bounds.
The proof is self-contained: it derives the Yamada-type difference set properties from character sums rather than citing them, so every step is checkable.
This does not resolve the full conjecture R(B_{n-1}, B_n) = 4n-1 for all n. In particular, the single-challenge case n = 50 from Epoch AI's FrontierMath benchmark sits in the q ≡ 7 (mod 8) branch, which this proof does not cover.
This was done as a human-AI collaboration using Claude and ChatGPT.
I will not trade on this market.
Links:
Draft proof, code, and data: https://github.com/baybar1507/book-ramsey-numbers
Epoch AI problem page: https://epoch.ai/frontiermath/open-problems/ramsey-book-graphs
Epoch AI problem PDF (full statement and references): https://epoch.ai/files/open-problems/ramsey-book-graphs.pdf
Wesley's paper (the framework this builds on): https://arxiv.org/abs/2410.03625
FrontierMath: Open Problems (about page): https://epoch.ai/frontiermath/open-problems/about
1,000