Set Systems and Families of Permutations with Small Traces

Abstract : We study the maximum size of a set system on $n$ elements whose trace on any $b$ elements has size at most $k$. We show that if for some $b \ge i \ge 0$ the shatter function $f_R$ of a set system $([n],R)$ satisfies $f_R(b) < 2^i(b-i+1)$ then $|R| = O(n^i)$; this generalizes Sauer's Lemma on the size of set systems with bounded VC-dimension. We use this bound to delineate the main growth rates for the same problem on families of permutations, where the trace corresponds to the inclusion for permutations. This is related to a question of Raz on families of permutations with bounded VC-dimension that generalizes the Stanley-Wilf conjecture on permutations with excluded patterns.
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[Research Report] RR-7154, INRIA. 2009, pp.14
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Contributeur : Xavier Goaoc <>
Soumis le : jeudi 17 décembre 2009 - 18:22:25
Dernière modification le : jeudi 12 avril 2018 - 01:53:50
Document(s) archivé(s) le : jeudi 23 septembre 2010 - 11:06:35


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  • HAL Id : inria-00441376, version 2
  • ARXIV : 0912.2979


Otfried Cheong, Xavier Goaoc, Cyril Nicaud. Set Systems and Families of Permutations with Small Traces. [Research Report] RR-7154, INRIA. 2009, pp.14. 〈inria-00441376v2〉



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