Directed Ramsey and anti-Ramsey schemes and the Flexible Atom Conjecture

Abstract

In this paper, we shed new light on the Flexible Atom Conjecture. We first give finite representation results for relation algebras [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text]. Prior to our paper, only [Formula: see text] and [Formula: see text] 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 [Formula: see text] requires at least 14 points, any square representation of [Formula: see text] requires at least 11 points, and any square representation of [Formula: see text] requires at least 12 points. Our technique adapts previous work of Alm et al. [Algebra Univ. (2022)], in that we examine the combinatorial structure induced by the flexible atom.