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Statement

$R(k,\ell)$ is the off-diagonal Ramsey number, 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 (p. 1). Theorem 1 (p. 1). For each fixed integer $k\ge2$ the ratio of consecutive values tends to one:

$$\lim_{\ell\to\infty}\frac{R(k,\ell+1)}{R(k,\ell)}=1.$$

The manuscript introduces it with "Answering a question of Erdős [3, p. 99]", the 1971 Oxford problem list, where Erdős writes of $f(l,n)$ (his name for $R(l,n)$) "I cannot even prove $\lim_{n=\infty}f(l,n+1)/f(l,n)=1$". The case $k=2$ is "immediate, since $R(2,\ell)=\ell$" (p. 2); the site's Problem 1014 asks for fixed $k\ge3$.

Source. OpenAI, On the ratio of $R(k,\ell)$ and $R(k,\ell+1)$, three-page manuscript hosted at cdn.openai.com (retrieved 2026-09-18; PDF metadata dated 22 April 2026); Theorem 1 on p. 1, proof on pp. 2--3, read on the page images and in the text layer. The abstract states "The proof is due to an internal model at OpenAI." No refereed publication, arXiv version or independent review was found on 2026-09-18; the site accepted the manuscript as the resolution on 24 April 2026. The card records the provenance.

Read depth. Claims checked: the statement, the definition and convention, and the statements of Lemmas 1--3 were read clause by clause on the page images. The one-page proof was read for its structure (below) and not checked step by step. Nothing here is independently reviewed.

Proof pointer

Section 2 (pp. 1--3; the proof of Theorem 1 is on pp. 2--3). Inputs: Lemma 1 (Erdős--Szekeres, $R(k,\ell)\le\binom{k+\ell-2}{k-1}$), Lemma 2 ($R(k,\ell)\gg_k(\ell/\log\ell)^{k/2}$ for fixed $k\ge3$, stated as a standard probabilistic bound without proof or citation) and Lemma 3 (dependent random choice, from Fox--Sudakov, Lemma 2.1, or Zhao, Theorem 1.7.5). For $k\ge3$ let $s=\lceil k/2\rceil$, $t=\lfloor k/2\rfloor$, $q=k^2$ and take a graph $G$ on $N=R(k,\ell+1)-1$ vertices with no $K_k$ and $\alpha(G)\le\ell$. Then $\delta(G)\ge R(k,\ell+1)-R(k,\ell)-1$ (display (1): the non-neighbors of a vertex span no $K_k$ and no independent $\ell$-set). Lemma 3 with $m=R(t,\ell+1)$ gives $U\subseteq V(G)$ in which every $s$-subset has at least $R(t,\ell+1)$ common neighbors and $|U|$ satisfies display (2); $G[U]$ has no $K_s$ (its common neighborhood would contain a $K_t$, making a $K_k$) and no independent $(\ell+1)$-set, so $|U|\le R(s,\ell+1)-1$. Display (3) combines the two bounds; Lemma 1 gives $R(s,\ell+1)\ll_k\ell^{s-1}$ and $R(t,\ell+1)\ll_k\ell^{t-1}$, Lemma 2 gives $N\gg_k\ell^{k/2-o(1)}$, so the right side of (3) is $o(1)$ by the balanced choice of $s,t$ and $q=k^2$; taking $q$-th roots, $(R(k,\ell+1)-R(k,\ell))/(R(k,\ell+1)-1)\to0$, which gives the ratio estimate.

Dependencies

Lemma 1 (Erdős and Szekeres 1935, equation (3) of that paper), Lemma 2 (an uncited probabilistic lower bound; the manuscript's context paragraph attributes the general lower bounds to Bohman and Keevash 2010) and Lemma 3 (Fox and Sudakov 2011). External premises are taken at statement level; none was checked here.

Bears on

• Problem 1014: the statement for fixed $k\ge3$ is the problem; the site's label PROVED (LEAN) rests on this manuscript and on Boris Alexeev's Lean formalization in plby/lean-proofs, announced in the thread on 23 April 2026; the card's second development, maokami/ramsey-ratio-lean, was created on 25 April 2026, after the relabel.
• Problem 544: the ratio limit alone says nothing about $R(3,k+1)-R(3,k)$; the quantitative Remark 1 is what the site's consequence uses.