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Claim. E. Li, On the Duke--Erdős--Rödl problem at the one-third threshold, arXiv:2606.06522v1 (2 June 2026, the claim's date), 20 pp., announced by its author on the site's discussion thread on 10 June 2026. As its abstract states, for an $n$-vertex graph $G$ with $e(G)\ge n^2/k$ and $k\le n^{1/3}$, $G$ contains a subgraph $H_8$ with $\Omega(n^2/k^2)$ edges in which every two distinct edges lie together on a cycle of length at most $8$ contained in $H_8$, and a subgraph $H_6$ with $\Omega(n^2/k^3)$ edges in which every two distinct edges lie together on a cycle of length at most $6$ contained in $H_6$. In the density $\rho=e(G)/n^2$ these are cores of $\Omega(\rho^2n^2)$ and $\Omega(\rho^3n^2)$ edges for every $\rho\ge n^{-1/3}$. The $H_8$ result is the second clause of Problem 584 for $\delta\ge n^{-1/3}$, extending Fox and Sudakov's range $\delta>n^{-1/5}$.

The preprint states three further results that settle nothing under the reading the problem page adopts. The $H_6$ result lacks the condition that adjacent edges lie on a $C_4$, so it is not the first clause; without that condition it re-proves, for $\rho\ge n^{-1/3}$, what Theorem 1 of Duke, Erdős and Rödl (1984) gives for every density above $n^{-1/2}$. Under the ambient-witness convention, with the witnessing cycles taken in $G$, it claims that every graph with at least $n^2/k$ edges and $k=o(n^{1/2})$ has $\Omega(n^2/k^3)$ selected edges whose pairs lie on cycles of $G$ of length at most $6$, with adjacent pairs on $C_4$'s of $G$; the problem page reads the cycles as lying inside $H_1$, as the sources do, so this result does not bear on the first clause, though under an ambient reading it would, if correct, combine with Fox and Sudakov's theorem to answer the sparse question yes. Finally, with the cycles inside the subgraph and adjacent pairs on a $C_4$ inside it, it claims that for every fixed $\beta\in[1/3,1/2)$ there are bipartite graphs with $\Theta_\beta(n^{2-\beta})$ edges in which every such subgraph has only $O_\beta(\rho^3n^2/(\log n)^2)$ edges, built from random cyclic shift-lifts of $K_{q,q}$; that obstruction is confined to $1/3\le\beta<1/2$ and excludes no smaller $c$.

Covers. The second clause ($H_2$) of Problem 584 for $\delta\ge n^{-1/3}$, that is $\delta=n^{-c}$ for every $0<c\le1/3$. Nothing for the first clause.

Depends on. Nothing in this wiki.

Standing. Claimed: an unrefereed preprint, described from its arXiv abstract; the site's thread gives it no review and the site labels the problem OPEN. The claim is partial and its range of the second clause contains the one Fox and Sudakov proved, so the problem's standing is unchanged by it.