The independence number of a subset of an abelian group

TL;DR

This paper investigates the maximum size of special subsets called t-independent and weakly t-independent within abelian groups, providing exact values and bounds, especially for cyclic groups, to understand their combinatorial structure.

Contribution

It introduces and analyzes the concepts of t-independence and weak t-independence in abelian groups, offering new bounds and exact values for their maximum sizes.

Findings

Derived exact values for maximum sizes of t-independent sets.

Established asymptotic bounds for weakly t-independent sets.

Focused analysis on cyclic groups ${ m Z}_n$.

Abstract

We call a subset $A$ of the (additive) abelian group $G$ {\it $t$-independent} if for all non-negative integers $h$ and $k$ with $h+k \leq t$, the sum of $h$ (not necessarily distinct) elements of $A$ does not equal the sum of $k$ (not necessarily distinct) elements of $A$ unless $h=k$ and the two sums contain the same terms in some order. A {\it weakly $t$-independent} set satisfies this property for sums of distinct terms. We give some exact values and asymptotic bounds for the size of a largest $t$-independent set and weakly $t$-independent set in abelian groups, particularly in the cyclic group ${\mathbb Z}_n$.