Loose Hamilton Cycles in Random 3-Uniform Hypergraphs

In the random hypergraph H = Hn,p;3 each possible triple appears independently with probability p. A loose Hamilton cycle can be described as a sequence of edges {xi , yi , xi+1} for i = 1, 2, . . . , n/2 where x1, x2, . . . , xn/2 , y1, y2, . . . , yn/2 are all distinct. We prove that there exists an absolute constant K > 0 such that if p ≥ K log n n2 then limn→∞ 4|n Pr(Hn,p;3 contains a loose Hamilton cycle) = 1.