Countable Homogeneous Tournaments

A tournament Tis called homogeneous just in case every isomorphism of subtournaments of smaller cardinality can be lifted to an automorphism of T. It is shown that there are precisely three homogeneous tournaments of power N(). Some analogous results for 2-tournaments are obtained. In this paper we characterize countable tournaments which are homogeneous in the sense of Fraissé [1]. As far as we know, interest in such a characterization dates from 1976 when Woodrow showed in his dissertation [10] that the only finite homogeneous tournaments have orders 1 and 3 and that up to isomorphism there are only two countably infinite homogeneous tournaments which do not embed the tournament D shown in Figure 1. The next year Woodrow and the author [7] characterized countable homogeneous graphs. This reawakened the author's interest in the tournament problem. Independently, Schmerl [9] characterized countable homogeneous partial orderings and also formulated the question as to which countable tournaments are homogeneous.