$d$-cluster-free sets with a given matching number

Let $3\le d\le k$ and $\nu \ge 0$ be fixed and $\mathcal{F}\subset \left(\genfrac{}{}{0px}{}{[n]}{k}\right)$. The matching number of $\mathcal{F}$, denoted by $\nu (\mathcal{F})$, is the maximum number of pairwise disjoint sets in $\mathcal{F}$, and $\mathcal{F}$ is $d$-cluster-free if it does not contain $d$ sets with the union of size at most $2k$ and empty intersection. In this paper, we give a lower bound and an upper bound for the maximum size of a $d$-cluster-free family with a matching number at least $\nu +1$. In particular, our result of the case $\nu =1$ settles a conjecture of Mammoliti and Britz. We also introduce a Tur'{a}n problem in hypergraphs that allows multiple edges, which may be of independent interest.