The Size of a Hypergraph and its Matching Number

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