Hereditary properties of partitions, ordered graphs and ordered hypergraphs

  • Authors:

  • József Balogh;Béla Bollobás;Robert Morris

  • Affiliations:

  • Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL;Department of Mathematical Sciences, The University of Memphis, Memphis, TN and Trinity College, Cambridge, United Kingdom;Department of Mathematical Sciences, The University of Memphis, Memphis, TN

  • Venue:
  • European Journal of Combinatorics - Special issue on extremal and probabilistic combinatorics
  • Year:
  • 2006

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Abstract

In this paper we use the Klazar-Marcus-Tardos method (see [A. Marcus, G. Tardos, Excluded permutation matrices and the Stanley-Wilf conjecture. J. Combin. Theory Ser. A 107 (2004) 153-160]) to prove that, if a hereditary property of partitions P has super-exponential speed, then, for every k- permutation π, P contains the partition of |2k| with parts {{i, π(i) + k} : i ∈ ⌈ k ⌉}. We also prove a similar jump, from exponential to factorial, in the possible speeds of monotone properties of ordered graphs, and of hereditary properties of ordered graphs not containing large complete, or complete bipartite ordered graphs.Our results generalize the Stanley-Wilf conjecture on the number of n-permutations avoiding a fixed permutation, which was recently proved by the combined results of Klazar [M. Klazar, The Füredi-Hajnal conjecture implies the Stanley-Wilf conjecture, in: D. Krob, A.A. Mikhalev, A.V. Mikhalev (Eds.), Formal Power Series and Algebraic Combinatorics, Springer, Berlin, 2000, pp. 250-255] and Marcus and Tardos [A. Marcus, G. Tardos, Excluded permutation matrices and the Stanley-Wilf conjecture, J. Combin. Theory Ser. A 107 (2004) 153-160]. Our main results follow from a generalization to ordered hypergraphs of the theorem of Marcus and Tardos.