A Dirac-type theorem for Hamilton Berge cycles in random hypergraphs

A Hamilton Berge cycle of a hypergraph on n vertices is an alternating sequence (v1,e1,v2,…,vn,en) of distinct vertices v1,…,vn and distinct hyperedges e1,…,en such that {v1,vn}⊆en and {vi,vi+1}⊆ei for every i∈[n−1]. We prove a Dirac-type theorem for Hamilton Berge cycles in random r-uniform hypergraphs by showing that for every integer r≥3 there exists k=k(r) such that for every γ>0 and p≥logk(r)⁡(n)nr−1 asymptotically almost surely every spanning subhypergraph H⊆H(r)(n,p) with minimum vertex degree δ1(H)≥(12r−1+γ)p(n−1r−1) contains a Hamilton Berge cycle. The minimum degree condition is asymptotically tight and the bound on p is optimal up to possibly the logarithmic factor. As a corollary this gives a new upper bound on the threshold of H(r)(n,p) with respect to Berge Hamiltonicity.