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Statement

Remark 1. "The proof gives a quantitative bound. For each fixed $k\ge2$, there is a constant $c_k>0$ such that

$$\frac{R(k,\ell+1)}{R(k,\ell)}\le1+\ell^{-c_k}$$

for all sufficiently large $\ell$. We do not attempt to optimize $c_k$." (p. 1, quoted as printed.)

Discrepancy of form, recorded here. The site's commentary on Problem 1014 and the formal-conjectures variant erdos_1014.variants.upper_bound print $R(k,\ell+1)\le(1+O(\ell^{-c/k^2}))R(k,\ell)$ with one constant $c>0$ for all $k$. The manuscript states only $c_k>0$ depending on $k$; the $k^2$ in the denominator is a reading of its proof, which takes a $q$-th root with $q=k^2$ of a quantity bounded by a fixed negative power of $\ell$. No value of $c_k$ is stated for any $k$, and nothing in the manuscript claims $c_3\ge1$.

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); Remark 1 on p. 1, read on the page image and in the text layer; the same provenance and attribution as Theorem 1.

Read depth. Claims checked: the remark was read clause by clause on the page image. Its justification is the last display of the proof of Theorem 1 (p. 2), read for structure and not checked.

Proof pointer

The proof of Theorem 1 (pp. 2--3): the right side of display (3) is bounded by fixed negative powers of $\ell$ through Lemmas 1--2, and the $q$-th root with $q=k^2$ gives the rate; the manuscript does not write the exponent out.

Dependencies

The proof of Theorem 1 and its three lemmas.

Bears on

• Problem 544: with $k=3$ (the manuscript's first argument) and the problem's $k$ as $\ell$, the remark reads $R(3,k+1)-R(3,k)\le k^{-c_3}R(3,k)$ for all large $k$, the site's consequence $R(3,k+1)-R(3,k)\ll k^{-c}R(3,k)$; an authored one-line specialization made on the problem page. It bounds the increment above by $O(k^{2-c_3}/\log k)$ and gives no lower bound, so it decides neither question of the problem unless $c_3\ge1$, which is not claimed.
• Problem 1014: the quantitative form the site's commentary reports, in the manuscript's own words.