TY - JOUR

T1 - Hereditary properties of partitions, ordered graphs and ordered hypergraphs

AU - Balogh, József

AU - Bollobás, Béla

AU - Morris, Robert

N1 - Funding Information:
The first author was supported during this research by OTKA grant T049398 and US National Science Foundation (NSF) grant DMS-0302804, the second by NSF grant ITR 0225610, and the third by a Van Vleet Memorial Doctoral Fellowship.

PY - 2006/11

Y1 - 2006/11

N2 - 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 [2 k] 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.

AB - 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 [2 k] 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.

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U2 - 10.1016/j.ejc.2006.05.004

DO - 10.1016/j.ejc.2006.05.004

M3 - Article

AN - SCOPUS:33748538660

SN - 0195-6698

VL - 27

SP - 1263

EP - 1281

JO - European Journal of Combinatorics

JF - European Journal of Combinatorics

IS - 8 SPEC. ISS.

ER -