Counting Hamilton Cycles in Dirac Hypergraphs

For 0≤ <k 0 ≤ ℓ < k , a Hamilton ℓ -cycle in a k -uniform hypergraph H is a cyclic ordering of the vertices of H in which the edges are segments of length k and every two consecutive edges overlap in exactly ℓ vertices. We show that for all 0≤ <k-1 0 ≤ ℓ < k - 1 , every k -graph with minimum co-degree δ n δ n with δ >1/2 δ > 1 / 2 has (asymptotically and up to a subexponential factor) at least as many Hamilton ℓ -cycles as a typical random k -graph with edge-probability δ δ . This significantly improves a recent result of Glock, Gould, Joos, Kühn and Osthus, and verifies a conjecture of Ferber, Krivelevich and Sudakov for all values 0≤ <k-1 0 ≤ ℓ < k - 1 .