A generalization of Erdos' matching conjecture

Let H = (V,epsilon ) be an r-uniform hypergraph on n vertices and fix a positive integer k such that 1 <= k <= r. A k-matching of H is a collection of edges M subset of epsilon such that every subset of V whose cardinality equals k is contained in at most one element of M. The k-matching number of H is the maximum cardinality of a k-matching. A well-known problem, posed by Erdos, 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 r is among this candidate set when n >= 4r ((r)(k) )(2). a.