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Claim. Theorem 1.1 of Conlon, Fox and Sudakov, quoted from p. 3 of the preprint read: "For $n$ sufficiently large, every $2$-coloring of the edges of the complete graph on the interval $\{2,\ldots,n\}$ contains a monochromatic clique with vertex set $S$ such that

$$\sum_{s\in S}\frac1{\log s}\ge2^{-8}\log\log\log n.$$

Hence, $f(n)=\Theta(\log\log\log n)$." Here $f(n)$ is the least, over all $2$-colorings of the pairs of $\{2,\ldots,n\}$, of the largest weight $\sum_{x\in X}1/\log x$ of a monochromatic clique $X$, and the upper half of the order is Rödl's coloring. The passage to the problem's question is one line: given $C>0$, every $n$ above the theorem's threshold with $2^{-8}\log\log\log n\ge C$ has a monochromatic $X$ of weight at least $C$. The theorem answers the question on its own, without Rödl's paper, and is paged at Theorem 1.1 of the library's source card, which cites arXiv v2 as the version read (16 October 2013, headed as accepted for Duke Mathematical Journal); v1 was posted on 7 December 2011.

Depends on. Nothing in this wiki; the theorem rests on the cited paper alone, and Rödl's coloring supplies only the upper half of the order, which is not the claim.

Acceptance. Refereed: Duke Math. J. 162 (2013), no. 15, 2903--2927 (the arXiv record's journal reference and the Crossref record); the journal text was not compared with the preprint read, so locators are preprint pages. Reviewed: the site's curator, T. F. Bloom, labels the problem PROVED (LEAN) and credits the paper, in the problem's commentary, with the theorem above and with showing Rödl's order best possible (page last edited 8 February 2026).

Read depth. Claims checked: Theorem 1.1, the definitions and the attributions of pp. 2--3 and Conjecture 5.1 with the remarks of p. 15 were read; the proof (Section 3, with the dependent random choice lemma of Section 2 and a weighted Ramsey theorem, built on Rödl's argument) was not read, and nothing is independently reviewed in this corpus. The constant in $f(n)$ is open (the paper's Conjecture 5.1) and is not this problem's question.