Multivalued generalizations of the Frankl--Pach Theorem

TL;DR

This paper extends the Frankl--Pach Theorem to multivalued systems using Gr"obner basis techniques, providing new bounds on the size of set systems that do not shatter certain subsets.

Contribution

It introduces two multivalued generalizations of the Frankl--Pach Theorem for n-tuple systems utilizing Gr"obner basis methods.

Findings

Established bounds for multivalued set systems

Applied Gr"obner basis techniques to combinatorial problems

Described standard monomials of Hamming spheres

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