Critical Subsets of Finite Abelian Groups

Sum-Free Sets

It is well-known that for a prime $p\equiv 2\pmod 3$ and integer $n\ge 1$, the maximum possible size of a sum-free subset of the elementary abelian group $\mathbb Z_p^n$ is $\frac13\,(p+1)p^{n-1}$. We establish a matching stability result in the case $p=5$: if $A\subseteq\mathbb Z_5^n$ is a sum-free subset of size $|A|>\frac32\cdot5^{n-1}$, then there are a subgroup $H<\mathbb Z_5^n$ of size $|H|=5^{n-1}$ and an element $e\notin H$ such that $A\subseteq(e+H)\cup(-e+H)$.

Sum-free sets in $Z_5^n$

In this thesis, we survey some results on sum-fiee sets of integers, sum-free sets in finite groups, and sum-free sequences, especially the reciprocal sum of the elements of a sum-free sequence. By studying locally maximal sum-free sets, we derive(known) bounds on Ramsey numbers R,( 3,2 ). Also some generalizations of Schur's theorem are discussed. We use .A group-theoretic and number-theoretic results in the thesis. As a final chapter, we present some open problems.

http://summit.sfu.ca/system/files/iritems1/7912/b15007194.pdf

Critical Subsets of Finite Abelian Groups

Citations

Sum-Free Sets

Sum-free sets in $Z_5^n$

Sum-free sets and sequences

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