Almost all triangle-free triple systems are tripartite

Abstract

A triangle in a triple system is a collection of three edges isomorphic to {123,124,345}. A triple system is triangle-free if it contains no three edges forming a triangle. It is tripartite if it has a vertex partition into three parts such that every edge has exactly one point in each part. It is easy to see that every tripartite triple system is triangle-free. We prove that almost all triangle-free triple systems with vertex set [n] are tripartite.

Our proof uses the hypergraph regularity lemma of Frankl and Rödl [13], and a stability theorem for triangle-free triple systems due to Keevash and the second author [15].

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Correspondence to József Balogh.

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Research supported in part by NSF CAREER Grant DMS-0745185 and DMS-0600303, UIUC Campus Research Board Grants 09072 and 08086, and OTKA Grant K76099.

Research supported in part by NSF grants DMS 0653946 and 0969092.

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Balogh, J., Mubayi, D. Almost all triangle-free triple systems are tripartite. Combinatorica 32, 143–169 (2012). https://doi.org/10.1007/s00493-012-2657-4

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Mathematics Subject Classification (2000)

  • 05C65
  • 05D40
  • 05C35
  • 05C30
  • 05A16