---

Claim. For every bipartite graph $G$ with $m$ edges and no isolated vertices,

$$r(G)\le2^{16\sqrt m+1}.$$

This is Theorem 5.2 of N. Alon, M. Krivelevich and B. Sudakov, Turán numbers of bipartite graphs and related Ramsey-type questions, Combin. Probab. Comput. 12 (2003), no. 5--6, 477--494, p. 487, paged as Theorem 5.2 of the library's source card. The half-page proof (p. 488) uses that $G$ is $\sqrt m$-degenerate and that the denser color class of a two-colored $K_n$, $n=2^{16\sqrt m+1}$, contains every $\sqrt m$-degenerate bipartite graph on $n^{1/4}>2m$ vertices (the paper's Theorem 3.6). The page is named by the issue, November 2003 according to its Crossref record, with the first day of the month standing in for the unknown day.

Covers. The question of Problem 546 for bipartite $G$, answered yes with $C=17$, since $16\sqrt m+1\le17\sqrt m$ for $m\ge1$. The paper's general bound $2^{7\sqrt m\log_2m}$ (Theorem 5.3, for all sufficiently large $m$) settles no instance and stays on the problem page.

Depends on. Nothing in this wiki; the theorem rests on the paper's Theorem 3.6.

Acceptance. Refereed: Combin. Probab. Comput. 12 (2003), no. 5--6, 477--494. Not reviewed: the site's PROVED label credits Sudakov's theorem, and the commentary's mention of this paper is not a review of it.