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Countable homogeneous tournaments. (English) Zbl 0562.05025
A tournament T is a pair (T,E) s.t. E is a binary relation over T with the properties (a) \(\forall \times \forall y(xEy\vee yEx\vee y=x)\) and (b) \(\forall x\forall y(xEy\to x\neq y\wedge -yEx)\). A countable tournament T is homogeneous if every isomorphism between finite subtournaments can be extended to an automorphism of T. Examples for homogeneous tournaments are the rationals with the strict order. The author proves that there are exactly five isomorphism types of countable (finite or infinite) homogeneous tournaments. This improves a result of Woodrow, who classified those countable homogeneous tournaments which omit a certain four-point tournament D. The main part of the paper is devoted to the proof that every homogeneous tournament T embedding D embeds every finite tournament. By a result of Fraissé, T is then unique up to isomorphism. The proof uses so called 2-tournaments. These are tournaments partioned into two classes. For 2-tournaments a general embedding property is proved. This gives the main theorem and a similar result for 2-tournaments which are countable and homogeneous.
Reviewer: K.Potthoff

MSC:
05C20 Directed graphs (digraphs), tournaments
03C50 Models with special properties (saturated, rigid, etc.)
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