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Almost all triangle-free triple systems are tripartite

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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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Authors and Affiliations

  1. University of Illinois, 1409 W. Green Street, Urbana, IL, 61801, USA

    József Balogh

  2. Department of Mathematics, U.C. California at San Diego, 9500 Gilmann Drive, La Jolla, USA

    József Balogh

  3. Department of Mathematics, Statistics, and Computer Science, University of Illinois, Chicago, IL, 60607, USA

    Dhruv Mubayi

Authors
  1. József Balogh
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  2. Dhruv Mubayi
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Corresponding author

Correspondence to József Balogh.

Additional information

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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  • DOI: https://doi.org/10.1007/s00493-012-2657-4

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