Multivalued Generalizations of the Frankl–pach Theorem

Abstract

In [13] P. Frankl and J. Pach proved the following uniform version of Sauer's Lemma.be an arbitrary d-uniform set system such that F does not shatter an s + 1-element set, then |F| ≤ n 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.

Optimal Multivalued Shattering

“…Theorem 5 [Hegedűs and Rónyai [9]] Let 0 ≤ d ≤ n and 0 ≤ d + s ≤ n + 1. Let C ⊆ (k) n be a code with no shattered set of size s and suppose that |{i ∈ [n] : c i = 0}| = d for every c ∈ C. Then…”