## Sphere packing numbers for subsets of the Boolean $$n$$-cube with bounded Vapnik-Chervonenkis dimension.(English)Zbl 0818.60005

Let $$V$$ be a subset of the Boolean $$n$$-cube $$\{0,1\}^ n$$ with Vapnik-Chervonenkis dimension $$d$$. Let $$M(k/n, V)$$ denote the cardinality of the largest subset $$W$$ of $$V$$ such that any two distinct vectors in $$W$$ differ on at least $$k$$ indices. It is shown that $M (k/n, V) \leq e(d + 1) \bigl( 2e (n + 1)/(k + 2d + 2) \bigr)^ d.$ This improves on the best previous result which contained an extra factor $$(\log (n/d))^ d$$. The new bound is best possible up to a multiplicative constant. There are applications in the theory of empirical processes.

### MSC:

 60C05 Combinatorial probability 68U05 Computer graphics; computational geometry (digital and algorithmic aspects) 05B40 Combinatorial aspects of packing and covering

### Keywords:

Vapnik-Chervonenkis dimension; empirical processes
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### References:

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