A Note on the Alon-Kleitman Argument for Sum-free Subset Theorem

TL;DR

This paper critiques the Alon-Kleitman argument for sum-free subsets in finite Abelian groups, clarifying its limited applicability and highlighting unresolved cases in the general setting.

Contribution

It identifies a mistake in the original argument and specifies its valid scope, contributing to the understanding of sum-free subset problems.

Findings

The original argument applies only to groups of the form (Z/pZ)^s.

The general case for arbitrary finite Abelian groups remains unresolved.

The paper clarifies the scope and limitations of the Alon-Kleitman argument.

Abstract

In 1990, Alon and Kleitman proposed an argument for the sum-free subset problem: every set of n nonzero elements of a finite Abelian group contains a sum-free subset A of size |A|>\frac{2}{7}n. In this note, we show that the argument confused two different randomness. It applies only to the finite Abelian group G = (Z/pZ)^s where p is a prime. For the general case, the problem remains open.