Small subgraphs of large minimum degree

Abstract

In 1991, Erdős, Faudree, Rousseau and Schelp asked to estimate the smallest $u=u(n,d,s)$ such that every $n$-vertex graph with average degree at least $d$ has a subgraph on at most $u$ vertices with minimum degree at least $s$. We resolve this problem up to polylogarithmic factors for all even values of $s$. More precisely, we prove that every $n$-vertex graph with minimum degree $d\geq s$ has a subgraph on $O_s\left(\max\left\{\frac{n}{d^{s/(s-2)}}(\log n)^{180},\log n\right\}\right)$ vertices with minimum degree at least $s$, whenever $s\ge 4$ is even. This improves a result of Janzer, Sudakov and Tomon, who obtained a similar conclusion for subgraphs of average degree at least $s$. We also prove that for every positive integer $s$ and every $\varepsilon>0$, there exists a finite family $\mathcal{H}$ of graphs with minimum degree $s$ such that $\operatorname{ex}(n,\mathcal{H})=O(n^{2-2/s+\varepsilon})$. This is tight and is the first result of similar kind that applies to odd values of $s$ greater than 3.