SUMMARY
Let H=(V,E)\mathcal{H}=(V,\mathcal{E}) be an rr-uniform hypergraph on nn vertices and fix a positive integer kk such that 1=k=r1\le k\le r. A kk-matching of H\mathcal{H} is a collection of edges M?E\mathcal{M}\subset \mathcal{E} such that every subset of VV whose cardinality equals kk is contained in at most one element of M\mathcal{M}. The kk-matching number of H\mathcal{H} is the maximum cardinality of a kk-matching. A well-known problem, posed by Erdos, asks for the maximum number of edges in an rr-uniform hypergraph under constraints on its 11-matching number. In this article we investigate the more general problem of determining the maximum number of edges in an rr-uniform hypergraph on nn vertices subject to the constraint that its kk-matching number is strictly less than aa. The problem can also be seen as a generalization of the well-known kk-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 an\ge 4r\binom{r}{k}^2\cdot a.