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The claim. Every graph on $n$ vertices with at least $C_kn\log n$ edges contains a $k$-regular subgraph (L. Pyber, Regular subgraphs of dense graphs, Combinatorica 5 (1985), no. 4, 347--349, the paper's main theorem as its zbMATH review, Zbl 0596.05035, states it). The constant is of order $k^2$: Erdős (Discrete Math. 72 (1988), p. 85) prints the result as $f_k(n)<c_2k^2n\log n$, and Chakraborti, Janzer, Methuku and Montgomery restate it as their Theorem 1.1, with average degree greater than $32r^2\log n$ forcing an $r$-regular subgraph in an $n$-vertex graph. Since $n\log n=n^{1+o(1)}$, the maximum asked for in Problem 182 is $\ll_k n^{1+o(1)}$ for every fixed $k\ge3$.

Covers. The yes-or-no part, answered yes; the order of the maximum is not determined. Erdős's 1988 survey presents the bound as the answer to the question he and Sauer could not settle, whether $f_3(n)<n^{1+\varepsilon}$, and Janzer and Sudakov (p. 2) record it as the result their Theorem 1.2 improves.

Read depth. The paper is not filed in the library; the statement is taken from its zbMATH review and from the restatements in the 1988 survey, paged at Section 6 of Erdős 1988, and in the paper of Chakraborti, Janzer, Methuku and Montgomery. The review says the proof rests on a lemma of Alon, Friedland and Kalai.

Acceptance. Refereed: Combinatorica 5 (1985), no. 4. The site's label PROVED credits Janzer and Sudakov, so this result is not reviewed.