Counting spanning subgraphs in dense hypergraphs

Abstract

We give a simple method to estimate the number of distinct copies of some classes of spanning subgraphs in hypergraphs with high minimum degree. In particular, for each $k\ge 2$ and $1\le \ell \le k-1$, we show that every $k$-graph on $n$ vertices with minimum codegree at least $$\cases{\left(\dfrac{1}{2}+o(1)\right)n & if $(k-\ell)\mid k$,\\ & \\ \left(\dfrac{1}{\lceil \frac{k}{k-\ell}\rceil(k-\ell)}+o(1)\right)n & if $(k-\ell)\nmid k$,}$$ contains $\exp (n\log n-\Theta (n))$ Hamilton $\ell$-cycles as long as $(k-\ell )\mid n$. When $(k-\ell )\mid k$ this gives a simple proof of a result of Glock, Gould, Joos, Kühn and Osthus, while, when $(k-\ell )\nmid k$ this gives a weaker count than that given by Ferber, Hardiman and Mond or, when $\ell <k/2$, by Ferber, Krivelevich and Sudakov, but one that holds for an asymptotically optimal minimum codegree bound.