Tur\'an numbers of hypergraph trees

Abstract

An $r$-graph is an $r$-uniform hypergraph tree (or $r$-tree) if its edges can be ordered as $E_1,\ldots, E_m$ such that $\forall i>1 \, \exists \alpha(i)<i$ such that $E_i\cap (\bigcup_{j=1}^{i-1} E_j)\subseteq E_{\alpha(i)}$. The Tur\'an number $ex(n,{\cal H})$ of an $r$-graph ${\cal H}$ is the largest size of an $n$-vertex $r$-graph that does not contain ${\cal H}$. A cross-cut of ${\cal H}$ is a set of vertices in ${\cal H}$ that contains exactly one vertex of each edge of ${\cal H}$. The cross-cut number $\sigma({\cal H})$ of ${\cal H}$ is the minimum size of a cross-cut of ${\cal H}$. We show that for a large family of $r$-graphs (largest within a certain scope) that are embeddable in $r$-trees, $ex(n,{\cal H})=(\sigma-1)\binom{n}{r-1}+o(n^{r-1})$ holds, and we establish structural stability of near extremal graphs. From stability, we establish exact results for some subfamilies.