# Comer Schemes, Relation Algebras, and the Flexible Atom Conjecture

## TL;DR

This paper explores relational structures from Comer schemes, introduces a new finite representation of a complex relation algebra, and uses SAT solvers to establish bounds on finite representability of specific algebras.

## Contribution

It generalizes Comer schemes, provides the first finite representation of a particular relation algebra, and applies SAT solvers to determine non-representability bounds.

## Key findings

- First finite representation of $34_{65}$.
- $33_{65}$ is not finitely representable on fewer than 24 points.
- $34_{65}$ is not representable on fewer than 24 points.

## Abstract

In this paper, we consider relational structures arising from Comer's finite field construction, where the cosets need not be sum free. These Comer schemes generalize the notion of a Ramsey scheme and may be of independent interest. As an application, we give the first finite representation of $34_{65}$. This leaves $33_{65}$ as the only remaining relation algebra in the family $N_{65}$ with a flexible atom that is not known to be finitely representable. Motivated by this, we complement our upper bounds with some lower bounds. Using a SAT solver, we show that $33_{65}$ is not finitely representable on fewer than $24$ points, and that $33_{65}$ does not admit a cyclic group representation on fewer than $120$ points. We also employ a SAT solver to show that $34_{65}$ is not representable on fewer than $24$ points.