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Claim. For every $r\ge2$ and every $n>r^2(r-1)/4$, every graph $G$ with $n$ vertices and $m\ge t_r(n)$ edges has a clique on $r$ vertices whose degree sum is at least $2rm/n$. The claimed result is R. J. Faudree, Complete subgraphs with large degree sums, J. Graph Theory 16 (1992), no. 4, 327--334, DOI 10.1002/jgt.3190160406. The paper is unread; the statement above is the one the introduction of Bollobás and Nikiforov's paper (p. 2; card) and the site's commentary give: "Later Faudree [7] proved the conjecture for any $r\ge2$ and $n>r^2(r-1)/4$." The publisher's abstract, as the Crossref record carries it, states the hypothesis as $n\ge k^2(k-1)/4$ and "$m<t(n,k)$" edges, the second a slip for $m\ge t(n,k)$, the direction the abstract's next sentence (the conclusion fails in general below $t(n,k)$) and the Bollobás--Nikiforov account give; it is recorded as printed. This proves the corrected Statement of Problem 904 on that range of $n$.

Covers. Every $r\ge2$ and every $n>r^2(r-1)/4$, with $m\ge t_r(n)$. The whole statement is the accepted full claim Bollobás--Nikiforov.

Depends on. Nothing in this wiki; the result rests on the cited paper alone.

Acceptance. Refereed: the Journal of Graph Theory, 16 (1992), no. 4. Reviewed: the site's curator, T. F. Bloom, credits Faudree with this range in the problem's commentary on a page labeled PROVED (LEAN), and Bollobás and Nikiforov's refereed paper credits it in its introduction.

Dating. The page is dated by the issue month, September 1992; the day is a placeholder.