Gupta 2024 optimizing cgms upper bound ramsey numbers

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corollary_21: The extension of the paper's explicit off-diagonal bound to Ramsey numbers with one red color and c further colors: the bound of Corollary 6, with ℓ the sum of the ℓ_i, times the multinomial-type factor Θ(ℓ), stated for all k and ℓ without the condition k ≥ ℓ.

corollary_6: The explicit off-diagonal bound that the paper's short inductive argument gives for all positive integers k ≥ ℓ, an improvement exponential in ℓ over the Erdős-Szekeres bound when log k ≪ ℓ < 0.6989k.

diagonal_bound_p3: The unnumbered diagonal specialization of Theorem 1 printed on p. 3, R(k,k) ≤ (3.7992…)^(k+o(k)), with its companion off-diagonal form R(k,ℓ) ≤ e^(−ℓ/20+o(k)) binom(k+ℓ, ℓ).

theorem_1: The optimized form of the book-algorithm bound on off-diagonal Ramsey numbers, uniform in 1 ≤ ℓ ≤ k, whose diagonal case is the 3.7992… bound.

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P. Gupta, N. Ndiaye, S. Norin and L. Wei, Optimizing the CGMS upper bound on Ramsey numbers. Preprint arXiv:2407.19026 (v1 26 July 2024; v2 29 August 2026). The arXiv listing carries no journal reference and a Crossref bibliographic query under the title returned no record (both read).

The copy read for this card is arXiv:2407.19026v2 [math.CO] 29 Aug 2026, 24 pages, with a text layer; printed page $=$ PDF page. Pages 1--3, 6, 20 and 23 were read on rendered page images, pages 5, 7, 8 and 19 on 2026-10-07, and pages 4 and 20--24 on 2026-10-08. The arXiv record names arXiv's non-exclusive distribution license (arXiv:2407.19026), every other right reserved.

Read status: claims checked for Theorem 1 and its error-term sentence (p. 2), the diagonal and off-diagonal displays on p. 3, the abstract's rounded bound (p. 1), Corollary 6 (p. 6), Remark 17, which closes Section 4 (pp. 19--20), and the declaration on p. 3, read clause by clause on the page images, with Fact 8 (p. 8) added on 2026-10-07 and Theorem 5 (p. 5), Theorem 20 (p. 21) and Corollary 21 (p. 22) on 2026-10-08; no proof was read beyond tracing which results the proof of Theorem 1 rests on.

Version and provenance, in the paper's own statements (recorded, not judged). The paper says (p. 3) that its main results were obtained in Summer 2024 without AI use; that the numerical calculations in the derivation of Theorem 1 from Theorem 14 "were incorrectly justified in the earliest public version" and were corrected in the second author's PhD thesis (McGill University, 2025, the paper's [Ndi25]); that a named AI model produced the present, shorter route from Theorem 14 to Theorem 1 "based on the outline provided by the authors", with the output "checked and edited by the authors, who take full responsibility for its correctness", and that the same model was used for proofreading; and that an AI coding tool was used to formalize Theorem 1 and Corollary 6 in Lean 4, at https://github.com/snorin239/RamseyLean. That repository was read statically at its head commit 90e87da214701dd6eb3d56a2c7121839d8269d14 (committed 2026-08-17T12:31:31Z; Lean and Mathlib pinned to v4.32.1). Its README and paper-to-Lean map name two public targets, RamseyLean.main for Theorem 1 (one error function $\eta$ with $\eta(k)=o(k)$, uniform in $\ell$, and the coefficients as the exact rationals $1/4$, $3/100$ and $2/25$) and RamseyLean.ramseyNumber_le_easy_optimized for Corollary 6; they state that the numerical optimization was redone with kernel-checked interval certificates rather than importing the manuscript's computer-algebra certificate, that the sources contain no sorry, admit or user-declared axioms, that #print axioms reports only propext, Classical.choice and Quot.sound for both targets, that Remark 17 (the unverified further optimization) and Section 5 are outside the formalization's scope, and that the formalization and its documentation were produced by an AI coding tool with minimal guidance from the paper's authors. The statement of RamseyLean.main was read as text and matches Theorem 1 up to the exact-rational coefficients. Nothing was built or audited here; the statements above are the repository's own.

The paper reworks the exponential improvement of the Erdős--Szekeres bound by Campos, Griffiths, Morris and Sahasrabudhe, replacing their book algorithm with a simple induction that maintains a lower bound on the excess of red edges between two vertex sets over a fixed density $p$. Corollary 6 (p. 6) gives the explicit bound $R(k,\ell)\le4(k+\ell)\bigl((\sqrt5+1)(k+2\ell)/(4\ell)\bigr)^\ell\bigl((k+2\ell)/k\bigr)^{k/2}$ for all positive integers $k\ge\ell$, "an improvement exponential in $\ell$ of the Erdős-Szekeres bound for $\log k\ll\ell<0.6989k$" (p. 6; the introduction rounds the threshold to $0.69k$), meaningful even when $\ell=o(k)$; Section 5 extends the induction to multicolor Ramsey numbers, where, by the authors' account (p. 2), extending the full book algorithm seems to face non-trivial technical obstacles. Theorem 1, the main result (p. 2), uses a higher-moment version of the maintained quantity and an optimized starting density to give $R(k,\ell)\le e^{G(\ell/k)k+o(k)}\binom{k+\ell}{\ell}$ for all positive integers $\ell\le k$, with $G(\lambda)=(-0.25\lambda+0.03\lambda^2+0.08\lambda^3)e^{-\lambda}$ and the $o(k)$ uniform in $1\le\ell\le k$. On the diagonal (p. 3) this reads $R(k,k)\le e^{-0.14e^{-1}k+o(k)}\binom{2k}{k}=(4e^{-0.14e^{-1}})^{k+o(k)}=(3.7992\ldots)^{k+o(k)}$, rounded to $(3.8)^{k+o(k)}$ in the abstract, and in general $R(k,\ell)\le e^{-\ell/20+o(k)}\binom{k+\ell}{\ell}$. Remark 17, which closes Section 4 (pp. 19--20), reports a "preliminary, unverified iteration" of the optimization, performed by an AI model at the authors' request, that would give the base $3.78233\ldots$ if verified, and the authors' expectation that lowering the base below $3.7$, "and, likely, even below $3.75$ would require new ideas". No new combinatorial ideas are claimed; the contribution is packaging and parameter optimization. The bearing on problem 77 is the diagonal bound, which gives $\limsup R(k)^{1/k}\le3.7992\ldots$.

Contents

• Introduction (pp. 1--3): the Erdős--Szekeres bound, the improvements of Thomason, Conlon and Sah, the two bounds (1) and (2) of Campos, Griffiths, Morris and Sahasrabudhe, display (3) (Corollary 6), the AI disclosure and the notation.
• Theorem 1 (p. 2): $R(k,\ell)\le e^{G(\ell/k)k+o(k)}\binom{k+\ell}{\ell}$ for all positive integers $\ell\le k$, the $o(k)$ uniform in $\ell$.
• Diagonal bound (p. 3): $R(k,k)\le(4e^{-0.14/e})^{k+o(k)}=(3.7992\ldots)^{k+o(k)}$ and $R(k,\ell)\le e^{-\ell/20+o(k)}\binom{k+\ell}{\ell}$.
• Corollary 6 (p. 6): the explicit short-proof bound above, from Theorem 5 (p. 5) with the density $p=((\sqrt5+1)k+(2\sqrt5-2)\ell)/((\sqrt5+1)(k+2\ell))$.
• Remark 17 (pp. 19--20), closing Section 4: the unverified further iteration and the authors' barrier expectation.
• Section 5 (pp. 20--22): the multicolor extension, Observation 18 (p. 20) through Theorem 20 (p. 21), Corollary 21 (p. 22), the multicolor form of Corollary 6 with the factor $\Theta(\boldsymbol\ell)=\ell^\ell/(\ell_1^{\ell_1}\cdots\ell_c^{\ell_c})$, and Remark 22 (p. 22) on the multicolor work of Balister et al.
• Appendix A (pp. 23--24): the interval-arithmetic data for Lemma 16.

Compiled scope

Pages 1--3, 6, 20 and 23 were read on the page images, pages 5, 7, 8 and 19 on 2026-10-07, and pages 4 and 20--24 on 2026-10-08; the proofs (Sections 2--5) were not read beyond tracing which results the proof of Theorem 1 rests on; the Lean repository was read as text at one commit and not built. Nothing here is independently reviewed.

Source: https://arxiv.org/abs/2407.19026.

Bears on. #77: the diagonal case of Theorem 1, $R(k,k)\le(3.7992\ldots)^{k+o(k)}$ (p. 3), gives $\limsup R(k)^{1/k}\le3.7992\ldots$, an upper bound on the limit should it exist; the existence and the value of the limit are untouched, and Remark 17, which is not a theorem, records the authors' expectation that lowering the base below $3.7$, and likely below $3.75$, would require new ideas. No problem page of this corpus cites Corollary 6 or Corollary 21; by a check made here, Corollary 6 at $\ell=k$ is weaker than $4^k$.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.