Abstract: | Let E = {X1, X2…, Xm} where the Xi ? V for 1 ≤ i ≤ m are distinct. The hypergraph G = (V, E) is said to be s-uniform if |X1| = s for 1 ≤ i ≤ m. A set of edges M = {Xi : i ? I } is a perfect matching if (i) i ≠ j ? I implies Xi ∩ Xi = 0, and (ii) ∪i?I Xi = V. In this article we consider the question of whether a random s-uniform hypergraph contains a perfect matching. Let s ≥ 3 be fixed and m/n4/3 → ∞. We show that an s-uniform hypergraph with m edges chosen uniformly from [74] contains a perfect matching with high probability. This improves an earlier result of Schmidt and Shamir who showed that m/n3/2 → ∞ suffices. © 1995 John Wiley & Sons, Inc. |