Ramsey properties for classes of relational systems
A class \({\mathcal K}\) of relational systems (of the same type) has the R- Ramsey property if for every \(S\in {\mathcal K}\) there is \(T\in {\mathcal K}\) such that to every splitting of the class of all relational subsystems of T isomorphic to R into two blocks, there is a subsystem U of T isomorphic to S with all the restrictions of U isomorphic to R falling in the same block. The authors show this holds when (a) \({\mathcal K}\) is a class of finite reflexive relation systems closed under products and \(R\in {\mathcal K}\) is a singleton; (b) \({\mathcal K}\) is a class of finite relational systems closed under isomorphism and with the strong amalgamation property, and \(R\in {\mathcal K}\) is such that for every \(S\in {\mathcal K}\) the restrictions of S isomorphic to R are pairwise disjoint. These results extend work of \textit{J. Nešetřil} [Eur. J. Comb. 4, 143-147 (1983; Zbl 0524.08008)]. Finally they show that for \({\mathcal K}\) an isomorphism closed class of finite reflexive or areflexive relational systems with the strong amalgamation property and \(R\in {\mathcal K}\), if \({\mathcal K}\) has the R-Ramsey property then R is constant (i.e. \(\pi (\rho)=\pi\) for all permutations \(\pi\) of the base set and all relations \(\rho\) in R).
scientific article; zbMATH DE number 3952820
scientific article