Explicit Constructions of Maximal 3-Zero-Sum-Free Subsets in $(\mathbb{Z}/4\mathbb{Z}{)}^n$

Abstract

We address a problem posed by Nathan Kaplan in the 2014 Combinatorial and Additive Number Theory session: finding the largest subset $H\subseteq (\mathbb{Z}/4\mathbb{Z}{)}^n$ with no distinct $x,y,z\in H$ such that $x+y+z\equiv 0(\mathrm{mod}4)$. For even-order abelian groups, a standard $|G|/2$ lower bound applies. We prove this is optimal for $G=(\mathbb{Z}/4\mathbb{Z}{)}^n$ using a pair-counting argument, with an explicit construction of vectors with first coordinate odd (1 or 3 mod 4), yielding size $2\times 4^{n-1}=4^n/2$ and density 0.5, verified for $n\le 10$. An AI-assisted hybrid greedy-genetic algorithm rediscovers this optimal size, highlighting its potential in combinatorial search.

3 pages, no figures, code available at https://github.com/DynMEP/ZeroSumFreeSets-Z4/releases/tag/v5.0.0