Small Subgraphs with Large Average Degree

In this paper we study the fundamental problem of finding small dense subgraphs in a given graph. For a real number \(s>2\), we prove that every graph on n vertices with average degree \(d\ge s\) contains a subgraph of average degree at least s on at most \(nd^{-\frac{s}{s-2}}(\log d)^{O_s(1)}\) vertices. This is optimal up to the polylogarithmic factor, and resolves a conjecture of Feige and Wagner. In addition, we show that every graph with n vertices and average degree at least \(n^{1-\frac{2}{s}+\varepsilon }\) contains a subgraph of average degree at least s on \(O_{\varepsilon ,s}(1)\) vertices, which is also optimal up to the constant hidden in the O(.) notation, and resolves a conjecture of Verstraëte.