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A fundamental problem of extremal graph theory is to ask, 'What is themaximum number of edges in an $F$-free graph on $n$ vertices?' Recently Alonand Shikhelman proposed a more general, subgraph counting, version of thisquestion. They considered the question of determining the maximum number ofcopies 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 isalso natural to ask what is the maximum number of copies of $T$ in an $F$-freegraph with $m$ edges and no restriction on the number of vertices. Frohmader,in a different context, determined the answer when $T$ and $F$ are bothcomplete graphs. We prove results for this problem analogous to theErd\H{o}s--Stone theorem, the Erd\H{o}s--Simonovits theorem, and the stabilitytheorem of Erd\H{o}s--Simonovits.

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