every tournament on vertices contains a transitive subtournament on vertices. The proof is simple: choose any one vertex to be part of this subtournament, and form the rest of the subtournament recursively on either the set of incoming neighbors of or the set of outgoing neighbors of , whichever is larger. For instance, every tournament on seven vertices contains a three-vertex transitive subtournament; the Paley tournament on seven vertices shows that this is the most that can be guaranteed . However, showed that this bound is not tight for some larger values of . proved that there are tournaments on vertices without a transitive subtournament of size Their proof uses a counting argument: the number of ways that a -element transitive tournament can occur as a subtournament of a larger tournament on labeled vertices is and when is larger than , this number is too small to allow for an occurrence of a transitive tournament within each of the different tournaments on the same set of labeled vertices. Paradoxical tournaments A player who wins all games would naturally be the tournament's winner. However, as the existence of non-transitive tournaments shows, there may not be such a player. A tournament for which every player loses at least one game is called a 1-paradoxical tournament. More generally, a tournament is called -paradoxical if for every -element subset of there is a vertex in such that for all . By means of the probabilistic method, Paul Erdős showed that for any fixed value of , if , then almost every tournament on is -paradoxical. On the other hand, an easy argument shows that any -paradoxical tournament must have at least players, which was improved to by Esther and George Szekeres (1965). There is an explicit construction of -paradoxical tournaments with players by Graham and Spencer (1971) namely the Paley tournament. Condensation The condensation of any tournament is itself a transitive tournament. Thus, even for tournaments that are not transitive, the strongly connected components of the tournament may be totally ordered. Score sequences and score sets The score sequence of a tournament is the nondecreasing sequence of outdegrees of the vertices of a tournament. The score set of a tournament is the set of integers that are the outdegrees of vertices in that tournament. Landau's Theorem (1953) A nondecreasing sequence of integers is a score sequence if and only if : Let be the