Chromatic graphs, Ramsey numbers and the flexible atom conjecture
Summary: 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 \(\mathcal M \subseteq L^3\). Let \(c: E \to L\). Let \([n]\) denote the integer set \(\{1,2,\dots,n\}\). For \(x,y,z \in V\), let \(c(xyz)\) denote the ordered triple \((c(xy),c(yz),c(xz))\). We say that \(c\) is good with respect to \(\mathcal M\) if the following conditions obtain: {\parindent=5mm \begin{itemize}\item[1.]\(\forall x,y \in V\) and \(\forall(c(xy),j,k) \in \mathcal M, \exists z \in V\) such that \(c(xyz)=(c(xy),j,k);\) \item[2.] \(\forall x,y,yz \in V, c(xyz) \in\mathcal M\); and \item[3.]\(\forall x \in V \forall \ell \in L \exists y \in V\) such that \(c(xy)=\ell\). \end{itemize}} We investigate particular subsets \({\mathcal M}\subseteq L^{3}\) and those edge colorings of \(K_{N}\) which are good with respect to these subsets \({\mathcal M}\). 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.
scientific article; zbMATH DE number 5540954