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

Abstract

Kalai conjectured that every $n$-vertex $r$-uniform hypergraph with more than $\frac{t-1}{r}\left(\genfrac{}{}{0ex}{}{n}{r-1}\right)$ 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 $\frac{t-1}{r}$ with the term $\frac{t-r+1}{r}$. We apply our main result in graphs to show an upper bound for the Turán number of trees.

References in corpus (1)

• Turán numbers of hypergraph trees