The Size of a Hypergraph and its Matching Number

More than forty years ago, Erdos conjectured that for any t <= n/k, every k-uniform hypergraph on n vertices without t disjoint edges has at most max{((kt-1)(k)), ((n)(k)) - ((n-t+1)(k))} edges. Although this appears to be a basic instance of the hypergraph Turan problem (with a t-edge matching as the excluded hypergraph), progress on this question has remained elusive. In this paper, we verify this conjecture for all t < n/3k(2). This improves upon the best previously known range t = O(n/k(3)), which dates back to the 1970s.