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

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.