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Iterating the Gupta–Ndiaye–Norin–Wei method: a computer-verified bound R(k)≤3.7698k+o(k)R(k)\le 3.7698^{k+o(k)}

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

Contributed by Ella Puerto ↗

Version 1 / Nov 03, 2026 / CC BY 4.0

Abstract

Gupta, Ndiaye, Norin and Wei (GNNW) recast the Campos–Griffiths–Morris–Sahasrabudhe method as an inductive theorem that turns a known bound on off-diagonal Ramsey numbers into a better one. With two applications they obtained R(k)≤(3.7992…)k+o(k)R(k)\le (3.7992\ldots)^{k+o(k)}. They also reported an unverified, AI-suggested third application giving 3.78233…3.78233\ldots. We make three observations about their framework. (i) The free function MM can be optimised pointwise. (ii) For a fixed input, the best output exponent is the least solution of an explicit first-order differential inequality. (iii) GNNW's Theorem 14 needs only F∈C1F\in C^1 with locally Lipschitz derivative. We then construct an explicit chain of eight applications and verify every hypothesis by interval arithmetic on the whole range 0<λ≤10<\lambda\le 1, with a hand-proved estimate covering λ→0\lambda\to0. The verification was done twice, with independent code, libraries and enclosure strategies. This proves R(k)≤3.76973k+o(k)R(k)\le 3.76973^{k+o(k)}, and in particular confirms GNNW's 3.782333.78233. Numerically, iterating the optimal step converges to about 3.76903.7690, which suggests this is close to the limit of the GNNW method in its present form.

Provenance statement

I did this work with extensive help from the AI assistant Claude (Anthropic), through Claude Code. Under my direction, Claude wrote the Python software, ran the parameter searches and interval-arithmetic checks, and drafted the manuscript. Two programs that share no code ran the verification independently: one uses mpmath intervals with mean-value forms, the other Arb balls with direct evaluation. Two auxiliary lemmas are also checked in Lean 4. I am responsible for the content. The code, certified data and Lean files are on Zenodo (doi:10.5281/zenodo.23885218).

References

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  3. M. Campos, S. Griffiths, R. Morris, J. Sahasrabudhe, An exponential improvement for diagonal Ramsey , Ann.\ of Math.\ 203 (2026), no. 3; arXiv:2303.09521.arXiv
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  7. P. Gupta, N. Ndiaye, S. Norin, L. Wei, Optimizing the CGMS upper bound on Ramsey numbers , arXiv:2407.19026v2.arXiv

Version history

  1. v1Submitted by Ella PuertoInitial depositCurrentNov 02, 2026