FINITELY CONSTRAINED CLASSES OF HOMOGENEOUS DIRECTED-GRAPHS

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 GAMMA. determines whether Q(GAMMA). the class of finite tournaments omitting every tournament in GAMMA, is well-quasi-order? First. we give a nonconstructive proof of the existence of an algorithm for the case in which GAMMA 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 GAMMA consists of two tournaments is also discussed.