On Triangle-Free Graphs Maximizing Embeddings of Bipartite Graphs

In 1991 Győri, Pach, and Simonovits proved that for any bipartite graph H containing a matching avoiding at most $1$ vertex, the maximum number of copies of H in any large enough triangle-free graph is achieved in a balanced complete bipartite graph. In this paper we improve their result by showing that if H is a bipartite graph containing a matching of size x and at most 1/2√x-1 unmatched vertices, then the maximum number of copies of H in any large enough triangle-free graph is achieved in a complete bipartite graph. We also prove that such a statement cannot hold if the number of unmatched vertices is Ω(x).