Supersaturated Sparse Graphs and Hypergraphs

A central problem in extremal graph theory is to estimate, for a given graph H, the number of H-free graphs on a given set of n vertices. In the case when H is not bipartite, Erd̋s, Frankl, and Rödl proved that there are 2(1+o(1))ex(n, H) such graphs. In the bipartite case, however, bounds of the form 2O(ex(n, H)) have been proven only for relatively few special graphs H. As a 1st attempt at addressing this problem in full generality, we show that such a bound follows merely from a rather natural assumption on the growth rate of n ↦ ex(n, H); an analogous statement remains true when H is a uniform hypergraph. Subsequently, we derive several new results, along with most previously known estimates, as simple corollaries of our theorem. At the heart of our proof lies a general supersaturation statement that extends the seminal work of Erd̋s and Simonovits. The bounds on the number of H-free hypergraphs are derived from it using the method of hypergraph containers.