Packing large balanced trees into bipartite graphs

Abstract

We prove that for every $\gamma >0$ there exists $n_0\in N$ such that for every $n\ge n_0$ any family of up to $n^{\frac{1}{2}-\gamma }$ trees having at most $(1-\gamma )n$ vertices in each bipartition class can be packed into $K_{n,n}$. As a tool for our proof, we show an approximate bipartite version of the Komlós–Sárközy–Szemerédi Theorem, which we believe to be of independent interest.