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Reports (Research Report) Year : 2009

## Set Systems and Families of Permutations with Small Traces

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Xavier Goaoc
Cyril Nicaud

#### 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.

### Dates and versions

inria-00441376 , version 1 (15-12-2009)
inria-00441376 , version 2 (17-12-2009)

### Identifiers

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

### Cite

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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