Finitely constrained classes of homogeneous directed graphs

Journal of Symbolic Logic 59 (1):124-139 (1994)
Given a finite relational language L is there an algorithm that, given two finite sets A and B of structures in the language, determines how many homogeneous L structures there are omitting every structure in B and embedding every structure in A? For directed graphs this question reduces to: Is there an algorithm that, given a finite set of tournaments Γ, determines whether QΓ, 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
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