  Let $\mathcal{H}=(V,\mathcal{E})$ be an $r$-uniform hypergraph on $n$
vertices and fix a positive integer $k$ such that $1\le k\le r$. A
$k$-\emph{matching} of $\mathcal{H}$ is a collection of edges
$\mathcal{M}\subset \mathcal{E}$ such that every subset of $V$ whose
cardinality equals $k$ is contained in at most one element of $\mathcal{M}$.
The $k$-matching number of $\mathcal{H}$ is the maximum cardinality of a
$k$-matching. A well-known problem, posed by Erd\H{o}s, asks for the maximum
number of edges in an $r$-uniform hypergraph under constraints on its
$1$-matching number. In this article we investigate the more general problem of
determining the maximum number of edges in an $r$-uniform hypergraph on $n$
vertices subject to the constraint that its $k$-matching number is strictly
less than $a$. The problem can also be seen as a generalization of the,
well-known, $k$-intersection problem. We propose candidate hypergraphs for the
solution of this problem, and show that the extremal hypergraph is among this
candidate set when $n\ge 4r\binom{r}{k}^2\cdot a$.
