HEXAGONbeta

The book Ramsey number R(B57,B58)=231R(B_{57},B_{58})=231

A mix of human-written and AI-generated textHuman understanding: some partsmath.CO — Combinatorics

Contributed by Deep Bhattacharjee ↗

Version 1 / Oct 22, 2026 / CC BY 4.0

Abstract

We construct a graph on 230230 vertices with no book B57B_{57} whose complement contains no B58B_{58}. With the upper bound of Rousseau and Sheehan this gives R(B57,B58)=231R(B_{57},B_{58})=231, the smallest case of Wesley's conjecture R(Bn−1,Bn)=4n−1R(B_{n-1},B_n)=4n-1 that was open. The graph is a two-block circulant graph over Z115\Z_{115}, and the proof reduces to 5757 inequalities.

Provenance statement

Deep Bhattacharjee directed the work. Claude (Anthropic), through Claude Code, wrote and ran the search that found the graph, wrote the independent C and Python checks and the Lean 4 formalisation, and drafted the text, which the author reviewed and edited. The result does not rest on any unchecked AI output: the paper proves it by hand through 57 inequalities over Z_115; two independently written programs recount the common neighbours and non-neighbours of all 26,335 vertex pairs; and the Lean 4 kernel checks the graph on every pair (decide +kernel, no native_decide) and proves the upper bound from Goodman's count, the final theorem depending only on propext, Classical.choice and Quot.sound. Programs, data and proofs: release v2.1.0, https://github.com/creelie/Breakthrough-problems-2/releases/tag/v2.1.0.
AI provenance filesAI-PROVENANCE-HEXAGON.md · 2 KB

Tools used

Anthropic
Claude CodeVersion Opus 5.5 Ultracode

References

  1. D. Bhattacharjee, Book Ramsey numbers R(B57,B58)=231R(B_{57},B_{58})=231 and R(B66,B67)=267R(B_{66},B_{67})=267: data, checks, search and Lean 4 proofs , release v2.1.0, 2026. https://github.com/creelie/Breakthrough-problems-2/releases/tag/v2.1.0link
  2. D. Bhattacharjee, A hand-checkable proof that R(B66,B67)=267R(B_{66},B_{67})=267 , preprint, 2026, archived in Ar .
  3. S. Chung, W. Du and W. J. Wesley, Autonomous mathematical discovery in an open-world multi-agent environment , preprint, 2026. arXiv:2608.23691 (https://arxiv.org/abs/2608.23691)arXiv
  4. S. Chung, W. Du and W. J. Wesley, Station data v2 , book Ramsey verification notebook and certificates, first public version 24 August 2026. https://github.com/dualverse-ai/station_data_v2, accessed 14 October 2026.link
  5. Epoch AI, FrontierMath open problems: Ramsey numbers for book graphs , 2026. https://epoch.ai/frontiermath/open-problems/ramsey-book-graphs, accessed 15 October 2026.link
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  7. L. de Moura and S. Ullrich, The Lean 4 theorem prover and programming language , in: Automated Deduction -- CADE 28, Lecture Notes in Comput. Sci. 12699 , Springer, Cham, 2021, 625--635. 10.1007/978-3-030-79876-5_37DOI
  8. P. Morris, The breakout method for escaping from local minima , in: Proceedings of the Eleventh National Conference on Artificial Intelligence (AAAI-93), AAAI Press, 1993, 40--45. https://cdn.aaai.org/AAAI/1993/AAAI93-007.pdflink
  9. C. C. Rousseau and J. Sheehan, On Ramsey numbers for books , J. Graph Theory 2 (1978), 77--87. 10.1002/jgt.3190020110DOI
  10. G. Szekeres, Tournaments and Hadamard matrices , Enseign. Math. (2) 15 (1969), 269--278.
Show all 12 referencesShow fewer references
  1. D. Turturean, Summary of new results on the Ramsey numbers for book graphs open problem , 2026. https://docs.google.com/document/d/1VinXOiMov2v-Y27GTJwoAECywv2a34MH4E8qa712zJYlink
  2. W. J. Wesley, Lower bounds for book Ramsey numbers , Discrete Math. 349 (2026), no. 5, 114913. 10.1016/j.disc.2025.114913DOI

Version history

  1. v1Submitted by Deep BhattacharjeeInitial depositCurrentOct 22, 2026