Sum-avoiding sets in groups

TL;DR

This paper investigates the structure of finite subsets in additive groups with bounded sum-avoiding subsets, providing a structural theorem and using nonstandard analysis to address a classical problem in additive combinatorics.

Contribution

It establishes a structure theorem for sum-avoiding sets in arbitrary additive groups, extending the Erd ext{"o}s-Moser problem to groups with torsion, and employs nonstandard analysis for the proof.

Findings

A structure theorem characterizing sum-avoiding sets in groups with torsion.

Negative answer to Erd ext{"o}s's question on large subsets with bounded sum-avoiding sets.

Positive results for groups whose order is not divisible by small primes.

Abstract

Let $A$ be a finite subset of an arbitrary additive group $G$, and let $\phi(A)$ denote the cardinality of the largest subset $B$ in $A$ that is sum-avoiding in $A$ (that is to say, $b_1+b_2 \not \in A$ for all distinct $b_1,b_2 \in B$). The question of controlling the size of $A$ in terms of $\phi(A)$ in the case when $G$ was torsion-free was posed by Erd\H{o}s and Moser. When $G$ has torsion, $A$ can be arbitrarily large for fixed $\phi(A)$ due to the presence of subgroups. Nevertheless, we provide a qualitative answer to an analogue of the Erd\H{o}s-Moser problem in this setting, by establishing a structure theorem, which roughly speaking asserts that $A$ is either efficiently covered by $\phi(A)$ finite subgroups of $G$, or by fewer than $\phi(A)$ finite subgroups of $G$ together with a residual set of bounded cardinality. In order to avoid a large number of nested inductive arguments, our proof uses the language of nonstandard analysis. We also answer negatively a question of Erd\H{o}s regarding large subsets $A$ of finite additive groups $G$ with $\phi(A)$ bounded, but give a positive result when $|G|$ is not divisible by small primes.