Directed Ramsey and Anti-Ramsey Schemes and the Flexible Atom Conjecture

TL;DR

This paper advances the understanding of the Flexible Atom Conjecture by providing finite representation results for specific relation algebras through generalized Ramsey schemes, and establishes new lower bounds on their representations.

Contribution

It introduces a generalized notion of directed Ramsey schemes for relation algebras and demonstrates finite representability for several algebras previously unknown to be finitely representable.

Findings

Finite representation results for multiple relation algebras.

Embedding of these algebras into finite directed (anti-)Ramsey schemes.

Lower bounds on the size of square representations for certain algebras.

Abstract

In this paper, we shed new light on the Flexible Atom Conjecture. We first give finite representation results for relation algebras $33_{37}, 35_{37}$, $77_{83}$, $78_{83}$, $80_{83}$, $82_{83}$, $83_{83}$, $1310_{1316}$, $1313_{1316}$, $1315_{1316}$, and $1316_{1316}$. Prior to our paper, only $83_{83}$ and $1316_{1316}$ were known to be finitely representable. We accomplish this by generalizing the notion of a relation algebra generated by a Ramsey scheme to the directed (antisymmetric) setting, and then showing that each of these algebras embeds into a finite directed (anti-)Ramsey scheme. The notion of a directed (anti-)Ramsey scheme may be of independent interest. We complement our upper bounds with some lower bounds. Namely, we show that any square representation of $31_{37}$ requires at least $14$ points, any square representation of $33_{37}$ requires at least $11$ points, and any square representation of $35_{37}$ requires at least $12$ points. Our technique adapts previous work of Alm, et. al. (Algebra Universalis 2022), in that we examine the combinatorial structure induced by the flexible atom.