Sum-avoiding sets in groups

Abstract

Let $A$ be a finite subset of an arbitrary additive group $G$, and let $\varphi (A)$ denote the cardinality of the largest subset $B$ in $A$ that is sum-avoiding in $A$ (that is to say, $b_1+b_2\notin A$ for all distinct $b_1,b_2\in B$). The question of controlling the size of $A$ in terms of $\varphi (A)$ in the case when $G$ was torsion-free was posed by Erdős and Moser. When $G$ has torsion, $A$ can be arbitrarily large for fixed $\varphi (A)$ due to the presence of subgroups. Nevertheless, we provide a qualitative answer to an analogue of the Erdős-Moser problem in this setting, by establishing a structure theorem, which roughly speaking asserts that $A$ is either efficiently covered by $\varphi (A)$ finite subgroups of $G$, or by fewer than $\varphi (A)$ 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 Erdős regarding large subsets $A$ of finite additive groups $G$ with $\varphi (A)$ bounded, but give a positive result when $|G|$ is not divisible by small primes.

27 pages, 2 figures. Formatted using the Discrete Analysis style file (this time with correct metadata)

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