Kalai’s conjecture in [formula omitted]-partite [formula omitted]-graphs

Abstract

Kalai conjectured that every n-vertex r-uniform hypergraph with more than t−1rnr−1 edges contains all tight r-trees of some fixed size t. We prove Kalai’s conjecture for r-uniform hypergraphs that are r-partite. Our result is asymptotically best possible up to replacing the term t−1r with the term t−r+1r. We apply our main result in graphs to show an upper bound for the Turán number of trees.