Sum-free cyclic multi-bases and constructions of Ramsey algebras

TL;DR

This paper explores the construction of Ramsey algebras by partitioning the non-identity elements of cyclic groups into symmetric sum-free cyclic bases, extending known results to all positive integers up to 400 except 8 and 13.

Contribution

It provides new constructions of Ramsey algebras for all positive integers up to 400, except for specific cases, advancing the understanding of their combinatorial structure.

Findings

Constructed Ramsey algebras for all m ≤ 400, except m=8 and m=13.

Extended previous results from m=2 to 7 to larger m values.

Demonstrated the existence of such structures in a broad range of cases.

Abstract

Given $X\subseteq \mathbb{Z}_N$, $X$ is called a \emph{cyclic basis} if $(X+X)\cup X=\mathbb{Z}_N$, \emph{symmetric} if $x\in X$ implies $-x \in X$, and \emph{sum-free} if $(X+X)\cap X=\varnothing$. We ask, for which $m$, $N\in\mathbb{Z}^+$ can the set of non-identity elements of $\mathbb{Z}_N$ be partitioned into $m$ symmetric sum-free cyclic bases? If, in addition, we require that distinct cyclic bases interact in a certain way, we get a proper relation algebra called a Ramsey algebra. Ramsey algebras (which have also been called Monk algebras) have been constructed previously for $2\leq m\leq 7$. In this manuscript, we provide constructions of Ramsey algebras for every positive integer $m$ with $2\leq m\leq 400$, with the exception of $m=8$ and $m=13$.