A note on hypergraphs without non-trivial intersecting subgraphs

Abstract

A hypergraph $\mathcal{F}$ is non-trivial intersecting if every two edges in it have a nonempty intersection but no vertex is contained in all edges of $\mathcal{F}$. Mubayi and Verstraëte showed that for every $k\ge d+1\ge 3$ and $n\ge (d+1)n/d$ every $k$-graph $\mathcal{H}$ on $n$ vertices without a non-trivial intersecting subgraph of size $d+1$ contains at most $\left(\genfrac{}{}{0ex}{}{n-1}{k-1}\right)$ edges. They conjectured that the same conclusion holds for all $d\ge k\ge 4$ and sufficiently large $n$. We confirm their conjecture by proving a stronger statement. They also conjectured that for $m\ge 4$ and sufficiently large $n$ the maximum size of a $3$-graph on $n$ vertices without a non-trivial intersecting subgraph of size $3m+1$ is achieved by certain Steiner systems. We give a construction with more edges showing that their conjecture is not true in general.

14 pages

References in corpus (1)

• The existence of designs II