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JCT
2011

Almost all triple systems with independent neighborhoods are semi-bipartite

9 years 4 months ago
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.
József Balogh, Dhruv Mubayi
Added 14 May 2011
Updated 14 May 2011
Type Journal
Year 2011
Where JCT
Authors József Balogh, Dhruv Mubayi
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