Turán numbers of Berge trees

Abstract

A classical conjecture of Erdős and Sós asks to determine the Turán number of a tree. We consider variants of this problem in the settings of hypergraphs and multi-hypergraphs. In particular, for all k and r, with r≥k(k−2), we show that any r-uniform hypergraph H with more than n(k−1)r+1 hyperedges contains a Berge copy of any tree with k edges different from the k-edge star. This bound is sharp when r+1 divides n and for such values of n we determine the extremal hypergraphs.