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Indivisible homogeneous directed graphs and a game for vertex partitions. (English summary) Discrete Math. 291 (2005), no. 1-3, 99–113. In 1972 Henson constructed the countable T-free homogeneous directed graphs HT (where T is a set of finite tournaments) which according to Cherlin (1998) make up the bulk of the countable homogeneous directed graphs. Such a graph is called divisible if there exists a partition of the vertices into two sets, neither of which contains an isomorphic copy of the graph. In the present paper it is proved that HT (for a possibly infinite set of tournaments) is indivisible if and only if for any two orbits X and Y (under stabilizers of finitely many points) X can be embedded into Y or Y can be embedded into X. For the sufficiency of the condition a game is used to define a partition of the orbits, given a partition of the vertices

Year: 2011

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