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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.)
##### Keywords:
homogeneous tournaments
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##### References:
 [1] Roland Fraïssé, Sur l’extension aux relations de quelques propriétés des ordres, Ann. Sci. Ecole Norm. Sup. (3) 71 (1954), 363 – 388 (French). · Zbl 0057.04206 [2] C. Ward Henson, A family of countable homogeneous graphs, Pacific J. Math. 38 (1971), 69 – 83. · Zbl 0204.24103 [3] C. Ward Henson, Countable homogeneous relational structures and ℵ$$_{0}$$-categorical theories, J. Symbolic Logic 37 (1972), 494 – 500. · Zbl 0259.02040 [4] Bjarni Jónsson, Universal relational systems, Math. Scand. 4 (1956), 193 – 208. · Zbl 0077.25302 [5] B. Jónsson, Homogeneous universal relational systems, Math. Scand. 8 (1960), 137 – 142. · Zbl 0173.00505 [6] A. H. Lachlan, Countable ultrahomogeneous tournaments, Abstracts Amer. Math. Soc. 1 (1980), 80T-A17. [7] A. H. Lachlan and Robert E. Woodrow, Countable ultrahomogeneous undirected graphs, Trans. Amer. Math. Soc. 262 (1980), no. 1, 51 – 94. · Zbl 0471.03025 [8] Michael Morley and Robert Vaught, Homogeneous universal models, Math. Scand. 11 (1962), 37 – 57. · Zbl 0112.00603 [9] James H. Schmerl, Countable homogeneous partially ordered sets, Algebra Universalis 9 (1979), no. 3, 317 – 321. · Zbl 0423.06002 [10] R. E. Woodrow, Theories with a finite number of countable models and a small language, Ph.D. Thesis, Simon Fraser Univ., Burnaby, British Columbia, Canada, 1976.
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