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 >= s contains a subgraph of average degree at least s on at most nd(-s/s-2) (log d)(Os(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(1-2/s+epsilon) contains a subgraph of average degree at least s on O-epsilon,O-s (1) vertices, which is also optimal up to the constant hidden in the O(.) notation, and resolves a conjecture of Verstraete.