[formula omitted]-cluster-free sets with a given matching number.

Let 3 ≤ d ≤ k and ν ≥ 0 be fixed. Let F ⊂ [ n ] k . The matching number of F , denoted by ν (F) , is the maximum number of pairwise disjoint sets in F. F is d -cluster-free if it does not contain d sets with union of size at most 2 k 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 ν + 1. In particular, our result of the case ν = 1 settles a conjecture of Mammoliti and Britz. We also introduce a Turán problem in hypergraphs that allows multiple edges, which may be of independent interest. [ABSTRACT FROM AUTHOR]