A note on hypergraphs without non-trivial intersecting subgraphs

TL;DR

This paper investigates the maximum size of hypergraphs avoiding certain intersecting subgraphs, confirming a conjecture for specific cases and providing counterexamples to a related conjecture.

Contribution

It proves a stronger version of Mubayi and Verstra"{e}te's conjecture for all relevant parameters and constructs hypergraphs that disprove another conjecture about Steiner systems.

Findings

Confirmed Mubayi and Verstra"{e}te's conjecture for all large n and relevant parameters.

Constructed hypergraphs with more edges than predicted by the disproved conjecture.

Provided theoretical bounds on the size of hypergraphs avoiding 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 Verstra\"{e}te 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 $\binom{n-1}{k-1}$ 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.