Small Dense Subgraphs of a Graph

Abstract

Given a family ${\cal F}$ of graphs, and a positive integer $n$, the Turan number $ex(n,{\cal F})$ of ${\cal F}$ is the maximum number of edges in an $n$-vertex graph that does not contain any member of ${\cal F}$ as a subgraph. The order of a graph is the number of vertices in it. In this paper, we study the Turan number of the family of graphs with bounded order and high average degree. For every real $d\geq 2$ and positive integer $m\geq 2$, let ${\cal F}_{d,m}$ denote the family of graphs on at most $m$ vertices that have average degree at least $d$. It follows from a simple application of the probabilistic method that $ex(n,{\cal F}_{d,m})=\Omega(n^{2-\frac{2}{d}+\frac{2}{dm}})$. Verstraete [personal communications] asked if it is true that for each fixed $d$ there exists a function $\epsilon_d(m)$ that tends to $0$ as $m\to \infty$ such that $ex(n,{\cal F}_{d,m})=O(n^{2-\frac{2}{d}+\epsilon_d(m)})$. We answer Verstraete's question in the affirmative whenever $d$ is an integer, showing that $ex(n,{\c...