Abundance of Unique Subhypergraphs

Abstract

Given $k$-uniform hypergraphs $G$ and $H$, we say that $G$ is a unique subhypergraph of $H$ if $H$ contains exactly one subhypergraph isomorphic to $G$. For an $n$-vertex $k$-graph $H$, let $f_k(H)$ be the number of non-isomorphic unique subhypergraphs of $H$, normalized by $2^{\left(\genfrac{}{}{0ex}{}{n}{k}\right)}/n!$, and let $f_k(n)$ be the maximum of $f_k(H)$ over all $n$-vertex $k$-graphs $H$. In the graph case $k=2$, Erdős asked whether there exists a constant $\hat{I}´>0$ such that $f_2(n)>\hat{I}´$ for all $n$, offering $100 for a proof and $25 for a disproof. Recently, Bradač and Christoph answered this question in the negative,, proving that $f_2(n)$ tends to $0$, or equivalently that no $n$-vertex graph contains a positive proportion of all $n$-vertex graphs as unique subgraphs. In this paper we show that the situation is fundamentally different for $k$-uniform hypergraphs with $k\ge 3$. In particular, for every fixed integer $k\ge 3$, we prove that ${\mathrm{liminf}}_{n\to \infty }{f}_k(n)\ge 2/9$.