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 ≥ log (n) nr−1 asymptotically almost surely every spanning subhypergraph H ⊆ H(r)(n, p) with minimum vertex degree δ1(H) ≥ ( 1 2r−1 + γ ) p ( n−1 r−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.