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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 $\mathbb{Z}_N$ bepartitioned into $m$ symmetric sum-free cyclic bases? If, in addition, werequire that distinct cyclic bases interact in a certain way, we get a properrelation algebra called a Ramsey algebra. Ramsey algebras (which have also beencalled Monk algebras) have been constructed previously for $2\leq m\leq 7$. Inthis manuscript, we provide constructions of Ramsey algebras for every positiveinteger $m$ with $2\leq m\leq 400$, with the exception of $m=8$ and $m=13$.

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