Openai 2026 ratio consecutive ramsey numbers

../

remark_1: The quantitative form of the consecutive-ratio theorem as the manuscript prints it, with an unspecified exponent depending on k; the site's displayed form with exponent c/k^2 is a reading of the proof.

theorem_1: The consecutive-ratio theorem for off-diagonal Ramsey numbers, proved by dependent random choice on a critical graph; the site's accepted resolution of Problem 1014, attributed to an internal model at OpenAI.

---

OpenAI, On the ratio of $R(k,\ell)$ and $R(k,\ell+1)$. A three-page manuscript hosted at https://cdn.openai.com/pdf/6dc7175d-d9e7-4b8d-96b8-48fe5798cd5b/Ramsey.pdf (2026). The text carries no author names, no date, no affiliation beyond the sentence quoted below, no arXiv identifier and no journal; the PDF metadata gives a creation date of 22 April 2026, and the site's Problem 1014 page accepted it on 24 April 2026 (the page's last edit) after a thread comment of 23 April 2026 linked it.

The copy read for this card is that file, three pages with a complete text layer, read on the rendered page images and in the text layer. Provenance: retrieved from the URL above (HTTP 200, one request); 74,985 bytes. No notice is printed on any of the file's three pages, and the publisher's terms-of-use page (https://openai.com/policies/terms-of-use/) could not be read (HTTP 403); the term is unstated.

Attribution as the manuscript states it: the abstract ends "The proof is due to an internal model at OpenAI." The card records that sentence as the source's own provenance and claims no independent check of the argument. This is a source-supported solution accepted by the site, distinct from a claim of journal refereeing: no refereed publication, no arXiv version and no independent review of the manuscript was found on 2026-09-18 (Crossref bibliographic query for the title, no record; the site's proof-claim tab for Problem 1014, empty). Two external Lean developments read statically at pinned commits are listed under Formal artifacts below; neither was built here.

Read status: claims checked for Theorem 1, Remark 1 and the statements of Lemmas 1--3 (read clause by clause on the page images of pp. 1--2); the one-page proof of Theorem 1 (pp. 2--3) was read for its structure and not checked step by step; nothing here is independently reviewed.

Contents

• Section 1, Background (p. 1): $R(k,\ell)$ is the least $N$ for which every graph on $N$ vertices has $k$ pairwise adjacent vertices or $\ell$ pairwise non-adjacent ones, with $R(1,\ell)=1$ by convention; "Answering a question of Erdős [3, p. 99]" (Erdős's 1971 Oxford problem list), Theorem 1: for each fixed integer $k\ge2$, $R(k,\ell+1)/R(k,\ell)\to1$ as $\ell\to\infty$. Remark 1: "For each fixed $k\ge2$, there is a constant $c_k>0$ such that $R(k,\ell+1)/R(k,\ell)\le1+\ell^{-c_k}$ for all sufficiently large $\ell$. We do not attempt to optimize $c_k$." The context paragraph cites the exponential improvement for diagonal Ramsey numbers of Campos, Griffiths, Morris and Sahasrabudhe [2] (and Gupta, Ndiaye, Norin and Wei [6]), Mattheus and Verstraete [7] for $R(4,\ell)=\ell^{3+o(1)}$, the Bohman--Keevash lower bound $R(k,\ell)\gg_k\ell^{(k+1)/2}(\log\ell)^{1/(k-2)-(k+1)/2}$ [1] and the Erdős--Szekeres upper bound $R(k,\ell)\ll_k\ell^{k-1}$ [4], and explains the idea: a ratio $R(k,\ell+1)/R(k,\ell)\ge1+\varepsilon$ would give every vertex of a critical graph for $R(k,\ell+1)$ degree at least about $\varepsilon R(k,\ell+1)$, and dependent random choice turns that density into a contradiction.
• Section 2, Proof (pp. 1--3): three external inputs, Lemma 1 (Erdős--Szekeres, $R(k,\ell)\le\binom{k+\ell-2}{k-1}$ for $k,\ell\ge2$), Lemma 2 ($R(k,\ell)\gg_k(\ell/\log\ell)^{k/2}$ for fixed $k\ge3$, "a standard application of the probabilistic method", stated without proof) and Lemma 3 (dependent random choice, cited to Fox--Sudakov [5, Lemma 2.1] and Zhao [8, Theorem 1.7.5]: for positive integers $q,s,m$, an $N$-vertex graph of average degree $d$ has a set $U$ in which every $s$-subset has at least $m$ common neighbors and $|U|\ge d^q/N^{q-1}-\binom Ns((m-1)/N)^q$). The proof of Theorem 1: the case $k=2$ is immediate from $R(2,\ell)=\ell$; for $k\ge3$ set $s=\lceil k/2\rceil$, $t=\lfloor k/2\rfloor$, $q=k^2$, take a graph $G$ on $R(k,\ell+1)-1$ vertices with no $K_k$ and $\alpha(G)\le\ell$, note $\delta(G)\ge R(k,\ell+1)-R(k,\ell)-1$ (display (1)), apply Lemma 3 with $m=R(t,\ell+1)$ to get $U$ with display (2), show $|U|\le R(s,\ell+1)-1$ (a $K_s$ in $U$ would have at least $R(t,\ell+1)$ common neighbors, hence a $K_t$ and a $K_k$), combine into display (3), bound the right side by $o(1)$ through Lemmas 1--2 and $\binom Ns\le N^s/s!$, and take $q$-th roots: $(R(k,\ell+1)-R(k,\ell))/(R(k,\ell+1)-1)\to0$.
• References [1]--[8] (p. 3): Bohman--Keevash 2010; Campos, Griffiths, Morris and Sahasrabudhe (Ann. of Math., to appear; arXiv:2303.09521); Erdős 1971; Erdős--Szekeres 1935; Fox--Sudakov 2011; Gupta, Ndiaye, Norin and Wei (arXiv:2407.19026); Mattheus--Verstraete, Ann. of Math. (2) 199 (2024), 919--941; Zhao 2023.

Compiled scope

The whole manuscript was read (three pages). Theorem 1, Remark 1 and the three lemma statements are compiled as statements with the proof pointer above; the proof was not reconstructed and no step was checked. Lemma 2 is used in the manuscript without proof or citation; Lemma 3 is quoted from the literature. The quantitative form the site and the formal-conjectures file print, $R(k,\ell+1)\le(1+O(\ell^{-c/k^2}))R(k,\ell)$, is not the manuscript's printed statement (Remark 1 gives $1+\ell^{-c_k}$ with an unspecified $c_k$ depending on $k$); it is a reading of the proof's $q$-th root with $q=k^2$ and is recorded on the Remark 1 page as a discrepancy of form.

Formal artifacts (read statically, not built)

• plby/lean-proofs, the repository named by the formal_proof attribute of the formal-conjectures file 1014.lean: head of main 8822f7ddef30fadbd92e1c6ab4ed897af356af5e (committer date 2026-09-15T20:59:44Z, read through the GitHub API). src/v4.29.1/ErdosProblems/Erdos1014.lean (148,292 bytes, 3,511 lines; last changed 2026-06-24) declares "a Lean formalization of a solution to Erdős Problem 1014", names the informal author as an internal model at OpenAI and the formal authors as an AI coding assistant and Boris Alexeev, imports Mathlib, defines ramseyNumber k l as the least n such that every SimpleGraph (Fin n) has a k-clique or an l-independent set, and proves erdos1014 (k : ℕ) (hk : 3 ≤ k) : Tendsto (fun l => (ramseyNumber k (l + 1) : ℝ) / ramseyNumber k l) atTop (𝓝 1) with no sorry and no axiom declaration; its closing comment records #print axioms as propext, Classical.choice, Quot.sound. The src/latest copy (Lean and Mathlib v4.33.0, 134,939 bytes, last changed 2026-08-24) imports a repository utility module and names the theorem erdos_1014; the index ErdosProblems/Erdos1014.md lists copies for five toolchains. The theorem's docstring notes a rate $C'(\log\ell)^A/\ell^c$ for the ratio minus one.
• maokami/ramsey-ratio-lean: head of main f44f24789d9ed422b6ac2bb3e69e523d311b40a0 (committer date 2026-04-29T07:23:46Z; repository created 2026-04-25; read), toolchain leanprover/lean4:v4.28.0-rc1 with Mathlib at the matching tag. RamseyRatio/Basic.lean defines ramsey k ℓ as sInf {N | HasRamseyProperty N k ℓ}; RamseyRatio/MainTheorem.lean (849 lines, no sorry) proves ramsey_ratio_quantitative (k : ℕ) (hk : 2 ≤ k) : ∃ c > (0 : ℝ), ∀ᶠ ℓ : ℕ in atTop, (R(k, ℓ + 1) : ℝ) / R(k, ℓ) ≤ 1 + (ℓ : ℝ) ^ (-c) (Remark 1) and ramsey_ratio_tendsto_one (k : ℕ) (hk : 2 ≤ k) : Tendsto (fun ℓ : ℕ => (R(k, ℓ + 1) : ℝ) / R(k, ℓ)) atTop (𝓝 1) (Theorem 1, with the manuscript's range $k\ge2$) and ends with #print axioms; its README states that the build reports only the three standard axioms and that the development reproduces this manuscript, a copy of which it bundles. Its rendered "proof tour" page was read as a web page on 2026-09-18.

Neither development was built, kernel-checked or audited here; their definitions of the Ramsey number differ from each other and from the formal-conjectures SimpleGraph.classicalRamsey, and no bridging statement was checked. These are static readings of theorem statements and declared axioms only.

Bears on. #1014: Theorem 1 is the problem's statement (the site asks for fixed $k\ge3$; the manuscript proves $k\ge2$), the site's accepted resolution. #544: Remark 1 with $k=3$ is the source of the site's consequence $R(3,k+1)-R(3,k)\ll k^{-c}R(3,k)$; it bounds the increment above and says nothing about divergence or $o(k)$.

Results.

• Theorem 1 (p. 1): for each fixed integer $k\ge2$, $R(k,\ell+1)/R(k,\ell)\to1$ as $\ell\to\infty$.
• Remark 1 (p. 1): with an exponent $c_k>0$ depending on the fixed $k\ge2$, $R(k,\ell+1)/R(k,\ell)\le1+\ell^{-c_k}$ once $\ell$ is large enough.

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