A Ramsey theorem for countable homogeneous directed graphs. (English summary) Combinatorics and algorithms (Hsin Chu/Kaohsiung, 2000). Discrete Math. 253 (2002), no. 1-3, 45–61. Let T be a finite set of finite tournaments and H = HT be the countably infinite homogeneous directed graph that does not embed any member of T. Let r be the smallest positive integer such that for every finite vertex coloring of H, H contains an induced copy of itself which uses at most r colors. The author proves a characterization of r in terms of orbits: The number r equals the largest number of elements in an antichain of the partial quasi-order of (subgraphs induced by) orbits of H under an embedding. A nonempty (and necessarily infinite) set X ⊂ V (H) is an orbit of H if there is a finite set S ⊂ V (H) such that X is a block of (V (H) � S)/∼, where x ∼ y iff x and y arrow and are arrowed by the same vertices of S
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