A Constrained version of Sauer's Lemma
Joel Ratsaby
In: Algorithms, Trees, Combinatorics and Probabilities (2004) Birkhausser Verlag , pp. 543-551. ISBN 0817671285

Abstract

Sauer's Lemma is extended to classes $\mH$ of binary-valued functions on $[n]=\{1, \ldots, n\}$ which have a margin less than or equal to $N$ on $x\in[n]$, where the margin $\mu_h(x)$ of a binary valued function $h$ at a point $x\in [n]$ is defined as the largest non-negative integer $a$ such that $h$ is constant on the interval $I_a(x) =[x-a, x+a] \subseteq [n]$. Estimates are obtained for the cardinality of classes of binary valued functions with a margin of at least $N$ on a sample $S\subseteq[n]$.

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