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{\bf Jeremy F. Alm, Roger D. Maddux and Jacob Manske}
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{\bf Chromatic Graphs, Ramsey Numbers and the Flexible Atom Conjecture}
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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 {\it good with
respect to} ${\cal M}$ if the following conditions obtain:

%\begin{enumerate}
\item{1.} $\forall x,y \in V$ and $\forall (c(xy),j,k) \in {\cal M}$, $\exists z \in V$ such that $c(xyz) = (c(xy),j,k)$; 
\item{2.} $\forall x,y,z \in V$, $c(xyz) \in {\cal M}$; and
\item{3.} $\forall x \in V \ \forall \ell\in L \ \exists \, y\in V$  such that $ c(xy)=\ell $.
%\end{enumerate}

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.

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