Finitely constrained classes of homogeneous directed graphs

Abstract

AbstractGiven a finite relational language L is there an algorithm that, given two finite sets and of structures in the language, determines how many homogeneous L structures there are omitting every structure in and embedding every structure in ?For directed graphs this question reduces to: Is there an algorithm that, given a finite set of tournaments Γ, determines whether Γ, the class of finite tournaments omitting every tournament in Γ. is well-quasi-order?First, we give a nonconstructive proof of the existence of an algorithm for the case in which Γ consists of one tournament. Then we determine explicitly the set of tournaments each of which does not have an antichain omitting it. Two antichains are exhibited and a summary is given of two structure theorems which allow the application of Kruskal's Tree Theorem. Detailed proofs of these structure theorems will be given elsewhere.The case in which Γ consists of two tournaments is also discussed.