The group of the countable universal graph

Abstract

Let C be a set with at least two, and at most ℵ 0 , members, and for any set X let [ X ] 2 denote the set of its 2-element subsets. If Γ is a countable set, and F c is a function from [Γ] 2 into C, then the structure Γ c = (Γ, F c ) 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 x εΓ–dom α such that (∀ y εdom α) F c { x, y } = α(y).