The Size of a Hypergraph and its Matching Number

More than forty years ago, Erdős conjectured that for any t ≤ nk , 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 Turán 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 3k2 . This improves upon the best previously known range t = O ( n k3 ) , which dates back to the 1970’s.