Journal of Symbolic Logic

Finitely Constrained Classes of Homogeneous Directed Graphs

Brenda J. Latka

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Given a finite relational language $L$ is there an algorithm that, given two finite sets $\mathscr{A}$ and $\mathscr{B}$ of structures in the language, determines how many homogeneous $L$ structures there are omitting every structure in $\mathscr{B}$ and embedding every structure in $\mathscr{A}$? For directed graphs this question reduces to: Is there an algorithm that, given a finite set of tournaments $\Gamma$, determines whether $\mathscr{Q}_\Gamma$, 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.

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J. Symbolic Logic, Volume 59, Issue 1 (1994), 124-139.

First available in Project Euclid: 6 July 2007

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Latka, Brenda J. Finitely Constrained Classes of Homogeneous Directed Graphs. J. Symbolic Logic 59 (1994), no. 1, 124--139.

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