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

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 $m=13$.