On Turán numbers for disconnected hypergraphs

Abstract

We introduce the following simpler variant of the Turán problem: Given integers $n>k>r\ge 2$ and $m\ge 1$, what is the smallest integer $t$ for which there exists an $r$-uniform hypergraph with $n$ vertices, $t$ edges and $m$ connected components such that any $k$-subset of the vertex set contains at least one edge? We prove some general estimates for this quantity and for its limit, normalized by $\left(\genfrac{}{}{0ex}{}{n}{r}\right)$, as $n\to \infty$. Moreover, we give a complete solution of the problem for the particular case when $k=5$, $r=3$ and $m\ge 2$.