
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 be a finite subset of an arbitrary additive group , and let denote the cardinality of the largest subset in that is sum-avoiding in (that is to say, for all distinct ). The question of controlling the size of in terms of in the case when was torsion-free was posed by Erd\H{o}s and Moser. When has torsion, can be arbitrarily large for fixed 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 is either efficiently covered by finite subgroups of , or by fewer than finite subgroups of together with a residual set of bounded cardinality. In order to avoid a large number of nested inductive…
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