Loose Hamilton Cycles in Random Uniform Hypergraphs

In the random k-uniform hypergraph H(n,p;k) of order n each possible k-tuple appears independently with probability p. A loose Hamilton cycle is a cycle of order n in which every pair of adjacent edges intersects in a single vertex. We prove that if pn(k-1)/log n tends to infinity with n then lim Pr(H(n,p;k) contains a loose Hamilton cycle) = 1. n ->infinity 2(k-1)vertical bar n This is asymptotically best possible.