Chromatic Graphs, Ramsey Numbers and the Flexible Atom Conjecture

Abstract

Let $K_{N}$ denote the complete graph on $N$ vertices with vertex set $V = V(K_{N})$ and edge set $E = E(K_{N})$. For $x,y \in V$, let $xy$ denote the edge between the two vertices $x$ and $y$. Let $L$ be any finite set and ${\cal M} \subseteq L^{3}$. Let $c : E \rightarrow L$. Let $[n]$ denote the integer set $\{1, 2, \ldots, n\}$. For $x,y,z \in V$, let $c(xyz)$ denote the ordered triple $\big(c(xy)$, $c(yz), c(xz)\big)$. We say that $c$ is good with respect to ${\cal M}$ if the following conditions obtain: 1. $\forall x,y \in V$ and $\forall (c(xy),j,k) \in {\cal M}$, $\exists z \in V$ such that $c(xyz) = (c(xy),j,k)$; 2. $\forall x,y,z \in V$, $c(xyz) \in {\cal M}$; and 3. $\forall x \in V \ \forall \ell\in L \ \exists \, y\in V$ such that $ c(xy)=\ell $. We investigate particular subsets ${\cal M}\subseteq L^{3}$ and those edge colorings of $K_{N}$ which are good with respect to these subsets ${\cal M}$. We also remark on the connections of these subsets and colorings to projective planes, Ramsey theory, and representations of relation algebras. In particular, we prove a special case of the flexible atom conjecture.