Conjecture on small regular subgraphs with prescribed average degree

Let $s\geq 3$ be an integer. For sufficiently large $n$, let $G$ be an $n$-vertex graph with average degree at least $d$, where $C\log\log n\leq d\leq n^{\frac{s-2}{s}}$. Regular-subgraph conjecture. There is a constant $C=C(s)$ such that $G$ contains an $s$-regular subgraph on at most $nd^{-\frac{s}{s-2}}(\log n)^C$ vertices. This strengthens the paper’s bound for subgraphs of average degree at least $s$ by requiring regularity. The source notes that the unrestricted existence case is the Erdős–Sauer problem, resolved by Janzer and Sudakov, while the full order bound remains conjectural.