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Claim. Theorem 1 of S. Mattheus and J. Verstraete, The asymptotics of $r(4,t)$, Ann. of Math. (2) 199 (2024), 919--941 (arXiv:2306.04007v5, p. 3): as $t\to\infty$,

$$r(4,t)=\Omega\Bigl(\frac{t^3}{\log^4t}\Bigr),$$

where $r(4,t)$ is the least $n$ such that every graph on $n$ vertices contains a clique of order $4$ or an independent set of order $t$. In the letters of Problem 986 this is $R(4,k)\gg k^3/(\log k)^4$, the statement at $s=4$ with $c(4)=4$. The proof modifies a graph built from Hermitian unitals at random so that it has no $K_4$ and counts its independent sets by the container method.

Covers. The case $s=4$ of the statement only.

Depends on. Theorem 1 of Mattheus and Verstraete, the result page of the cited paper.

Postings. The first arXiv version was posted on 6 June 2023, the date this page is named by; v5 of 20 February 2024 is marked on arXiv as the updated journal version. The journal article was published online on 5 March 2024.

Acceptance. Refereed: Annals of Mathematics (2) 199 (2024), no. 2, 919--941. The site's commentary credits Mattheus and Verstraete with the case $s=4$, but its PROVED label settles the problem through Bradač's claim, so the curator's credit is not listed as review of this one.