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) 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 1-2 s +ε contains a subgraph of average degree at least s on O ε,s (1) vertices, which is also optimal up to the constant hidden in the O(.) notation, and resolves a conjecture of Verstraëte.