Multivalued generalizations of the Frankl--Pach Theorem

Details Multivalued generalizations of the Frankl--Pach Theorem Multivalued generalizations of the Frankl--Pach Theorem

2010-08-27

arxiv.org/abs/1008.4660v2

Mathematics

Combinatorics

21 pages, accepted by J. of Algebra and its Applications

Scientific paper

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 Gr\"obner basis methods, and describe the standard monomials of Hamming spheres.

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