Almost all triple systems with independent neighborhoods are semi-bipartite

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 H is semi-bipartite if its vertex set contains a vertex subset X such that every edge of H intersects X in exactly two points. It is easy to see that if H is semi-bipartite, then the neighborhood of every pair of vertices in H 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˝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.

• Type: Journal
• Year: 2011
• Where: JCT
• Authors: József Balogh, Dhruv Mubayi
• Keywords: Hypergraph Regularity Lemma, JCT 2011, Triple Systems, Vertex Subset