Stability and Erd\H{o}s--Stone type results for $F$-free graphs with a fixed number of edges

Abstract

A fundamental problem of extremal graph theory is to ask, 'What is the maximum number of edges in an $F$-free graph on $n$ vertices?' Recently Alon and Shikhelman proposed a more general, subgraph counting, version of this question. They considered the question of determining the maximum number of copies of a fixed graph $T$ in an $F$-free graph on $n$ vertices. In this more general context, where we are no longer counting edges, it is also natural to ask what is the maximum number of copies of $T$ in an $F$-free graph with $m$ edges and no restriction on the number of vertices. Frohmader, in a different context, determined the answer when $T$ and $F$ are both complete graphs. We prove results for this problem analogous to the Erd\H{o}s--Stone theorem, the Erd\H{o}s--Simonovits theorem, and the stability theorem of Erd\H{o}s--Simonovits.