Weighted EGZ-constant for p-groups of rank 2

Abstract

Let [Formula: see text] be a finite abelian group of exponent [Formula: see text], written additively, and let [Formula: see text] be a subset of [Formula: see text]. The constant [Formula: see text] is defined as the smallest integer [Formula: see text] such that any sequence over [Formula: see text] of length at least [Formula: see text] has an [Formula: see text]-weighted zero-sum subsequence of length [Formula: see text] and [Formula: see text] defined as the smallest integer [Formula: see text] such that any sequence over [Formula: see text] of length at least [Formula: see text] has an [Formula: see text]-weighted zero-sum subsequence of length at most [Formula: see text]. Here, we prove that, for [Formula: see text], and [Formula: see text], we have [Formula: see text] and classify all the extremal [Formula: see text]-weighted zero-sum free sequences.