The group of the countable universal graph

Abstract

Let C be a set with at least two, and at most ℵ0, members, and for any set X let [X]2 denote the set of its 2-element subsets. If Γ is a countable set, and Fc is a function from [Γ]2 into C, then the structure Γc = (Γ, Fc) is called the countable universal C-coloured graph if the following condition is satisfied:Whenever α is a map from a finite subset of Γ into C there is xεΓ–dom α such that (∀yεdom α) Fc {x, y} = α(y).