On Tur\'an numbers for disconnected hypergraphs

TL;DR

This paper studies a simplified Turán problem for hypergraphs, determining the minimal number of edges needed to ensure every k-vertex subset contains an edge, with specific solutions for certain parameters.

Contribution

It introduces a new variant of the Turán problem for disconnected hypergraphs and provides general bounds and exact solutions for specific cases.

Findings

Established bounds for the minimal number of edges in the new Turán variant.

Provided a complete solution for the case k=5, r=3, m≥2.

Analyzed the asymptotic behavior as the number of vertices grows.

Abstract

We introduce the following simpler variant of the Tur\'an problem: Given integers $n>k>r\geq 2$ and $m\geq 1$, 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 $\binom{n}{r}$, as $n\rightarrow \infty$. Moreover, we give a complete solution of the problem for the particular case when $k=5$, $r=3$ and $m\geq 2$.