Kalai's conjecture in $r$-partite $r$-graphs

Abstract

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