A note on hypergraphs without non-trivial intersecting subgraphs

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 $\left(\genfrac{}{}{0px}{}{n-1}{k-1}\right)$ 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.