Hereditary Properties of Triple Systems

Abstract

For an integer $s\ges 2$, a property $\P^{(s)}$ is an infinite class of s-uniform hypergraphs closed under isomorphism. We say that a property $\P^{(s)}$ is \emph{hereditary\/} if~$\P^{(s)}$ is closed under taking induced subhypergraphs. Thus, for some `forbidden class' $\FF=\{\F_i^{(s)}\:i\in I\}$ of s-uniform hypergraphs, $\P^{(s)}$ is the set of all s-uniform hypergraphs not containing any $\F_i^{(s)}\in\FF$ as an induced subhypergraph. Let $\P^{(s)}_n$ be those hypergraphs of $\P^{(s)}$ on some fixed n-vertex set. For a set of s-uniform hypergraphs $\FF=\{\F_i^{(s)}\:i\in I\}$, let

\[ \exind(n,\FF)=\max\bigl|[n]^s{\setminus}(\M\cup\N)\vphantom{\big|}\bigr|, \]

where the maximum is taken over all $\M$ and $\N\subseteq[n]^s$ with $\M\cap\N=\emptyset$ such that, for all $\G\subseteq[n]^s{\setminus}(\M\cup\N)$, no $\F_i^{(s)}\in\FF$ appears as an induced subhypergraph of $\G\cup\M$. We show that

\[ \log_2\big|\P^{(3)}_n\big|=\exind(n,\FF)+o(n^3) \]

holds for $s=3$ and any hereditary property $\P^{(3)}$, where $\FF$ is a forbidden class associated with $\P^{(3)}$. This result complements a collection of analogous theorems already proved for graphs (i.e., $s=2$).