The Size of a Hypergraph and its Matching Number

More than forty years ago, Erdős conjectured that for any t ≤ n/k , every k -uniform hypergraph on n vertices without t disjoint edges has at most max kt-1k, nk-n-t+1k\ 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 1970s.