Asymptotic Enumeration of $F$-decompositions in Typical Hypergraphs and of Universal Cycles

Source abstract

In this paper, we determine the asymptotic number of $F$-decompositions in typical hypergraphs. More precisely, let $F$ be a non-trivial $f$-vertex $k$-graph. We show that, for all sufficiently large $n$, every $n$-vertex $F$-divisible $(n^{-ζ},h,p)$-typical $k$-graph $G$ has $n^{\frac{f-k+o_n(1)}{|F|}|G|}$ $F$-decompositions. As an application, we asymptotically enumerate universal cycles for $k$-subsets of $[n]$, or equivalently, tight Euler tours of the complete $k$-graph $K_n^k$. For every $k\ge2$ and all sufficiently large $n$ satisfying $k\mid {n-1 \choose k-1}$, the number of such universal cycles is $n^{(1-γ(n)){n \choose k}}$, where $0<γ(n)=o_n(1)$. More generally, for $\ell\in[k-1]$, we show that a sufficiently typical $n$-vertex $k$-graph $G$ satisfying the corresponding divisibility condition has $n^{(k-\ell-γ'(n))|G|}$ Euler $\ell$-tours, where $0<γ'(n)=o_n(1)$.