Almost all triple systems with independent neighborhoods are semi-bipartite

Abstract: The neighborhood of a pair of vertices $u,v$ in a triple system is the set of vertices $w$ such that $uvw$ is an edge. A triple system $\HH$ is semi-bipartite if its vertex set contains a vertex subset $X$ such that every edge of $\HH$ intersects $X$ in exactly two points. It is easy to see that if $\HH$ is semi-bipartite, then the neighborhood of every pair of vertices in $\HH$ is an independent set. We show a partial converse of this statement by proving that almost all triple systems with vertex sets $[n]$ and independent neighborhoods are semi-bipartite. Our result can be viewed as an extension of the Erd\H os-Kleitman-Rothschild theorem to triple systems. The proof uses the Frankl-R\"odl hypergraph regularity lemma, and stability theorems. Similar results have recently been proved for hypergraphs with various other local constraints.