Kalai’s conjecture in r-partite r-graphs

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-uniform hypergraphs that are r-partite. 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.