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arXiv · 1008.4660

Multivalued generalizations of the Frankl--Pach Theorem

Gabor Hegedüs·Lajos Ronyai
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Abstract

P. Frankl and J. Pach proved the following uniform version of Sauer's Lemma. Let $n,d,s$ be natural numbers such that $d\leq n$, $s+1\leq n/2$. Let $\cF \subseteq {[n] \choose d}$ be an arbitrary $d$-uniform set system such that $\cF$ does not shatter an $s+1$-element set, then $$ |\cF|\leq {n \choose s}.$$ We prove here two generalizations of the above theorem to $n$-tuple systems. To obtain these results, we use Gröbner basis methods, and describe the standard monomials of Hamming spheres.

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Gabor Hegedüs, Lajos Ronyai. 2011-02-15. Multivalued generalizations of the Frankl--Pach Theorem. https://arxiv.org/abs/1008.4660

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