The least non-partitionable zero-sum subset for zero-sum triples in finite abelian groups

Abstract

Let \(G\) be a finite abelian group, and let \(μ(G)\) denote the least size of a subset \(S\subseteq G\) with \(3\mid |S|\), total sum zero, and no partition into zero-sum triples; put \(μ(G)=\infty\) if no such subset exists. We prove the exact classification \(μ(G)=\infty\) precisely for groups of order at most \(8\) and for \(G\cong C_3^2\), while \(μ(G)=6\) for every other finite abelian group. The cyclic special case gives \(μ(\mathbb Z_n)=\infty\) for \(1\le n\le 8\) and \(μ(\mathbb Z_n)=6\) for \(n\ge 9\), answering the corresponding non-partite cyclic question. We also record a higher-uniformity interval construction which explains the large cyclic witnesses as the case \(k=3\) of a general \(k\)-tuple obstruction.