---

Claim. Theorem 1.2 of J. Fox and B. Sudakov, On a problem of Duke--Erdős--Rödl on cycle-connected subgraphs, J. Combin. Theory Ser. B 98 (2008), no. 5, 1056--1062 (p. 1057): for $0<\beta<1/5$ and $n$ sufficiently large, every graph with $n$ vertices and at least $n^{2-\beta}$ edges has a strongly $C_8$-connected subgraph with at least $\tfrac1{64}n^{2-2\beta}$ edges, that is, a subgraph in which every two edges lie on a cycle of length at most $8$ inside it and every two edges sharing a vertex on one of length at most $6$. The authors say this settles their Problem 1.1, the question Duke, Erdős and Rödl posed in 1984, in its strengthened form. The claim's date is the first arXiv posting, arXiv:0706.1920v1 of 13 June 2007; the journal received the paper on 10 April 2007 and published it online on 8 February 2008. The theorem is recorded on the theorem page of the source card.

Covers. The second clause ($H_2$) of Problem 584 for $\delta=n^{-c}$, every $0<c<1/5$, with the absolute constant $1/64$ (adjacent pairs even lie on cycles of length at most $6$). Nothing for the first clause, so the question whether some $c>0$ makes both clauses hold stays open.

Depends on. Nothing in this wiki.

Acceptance. Refereed: the paper is a publication in the Journal of Combinatorial Theory, Series B. The site's commentary credits the paper with the second statement for $\delta>n^{-1/5}$ while labeling the problem OPEN, which is commentary on an open problem and not reviewed evidence.