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 $x + y + z \equiv 0 \pmod{4}$. 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 $n \leq 10$. An AI-assisted hybrid greedy-genetic algorithm rediscovers this optimal size, highlighting its potential in combinatorial search.