Equivalently, a tournament is an orientation of an undirected complete graph.

[1]) Equivalently, a tournament is a complete asymmetric relation.

Many of the important properties of tournaments were investigated by H. G. Landau in 1953 to model dominance relations in flocks of chickens.

[4] Tournaments are also heavily studied in voting theory, where they can represent partial information about voter preferences among multiple candidates, and are central to the definition of Condorcet methods.

This argument also gives an algorithm for finding the Hamiltonian path.

The Hamiltonian paths are in one-to-one correspondence with the minimal feedback arc sets of the tournament.

in the range from three to the number of vertices in the tournament, there is a cycle of length

[10] This result was extended by Bang-Jensen, Gutin & Yeo (1997).

In other words, in a transitive tournament, the vertices may be (strictly) totally ordered by the edge relation, and the edge relation is the same as reachability.

vertices: Transitive tournaments play a role in Ramsey theory analogous to that of cliques in undirected graphs.

The proof is simple: choose any one vertex

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.

[12] However, Reid & Parker (1970) showed that this bound is not tight for some larger values of

[13] Erdős & Moser (1964) 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

, this number is too small to allow for an occurrence of a transitive tournament within each of the

[12] 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.

By means of the probabilistic method, Paul Erdős showed that for any fixed value of

players by Graham and Spencer (1971) namely the Paley tournament.

Landau's Theorem (1953) A nondecreasing sequence of integers

(sequence A000571 in the OEIS) starts as: 1, 1, 1, 2, 4, 9, 22, 59, 167, 490, 1486, 4639, 14805, 48107, ... Winston and Kleitman proved that for sufficiently large n: where

Takács later showed, using some reasonable but unproven assumptions, that where

[18] In social choice theory, tournaments naturally arise as majority relations of preference profiles.

of voters is odd, then the majority relation forms the dominance relation of a tournament on vertex set

vertices can be obtained as the majority relation of at most

[20] Results by Stearns and Erdős & Moser later established that

voters are needed to induce every tournament on

[22] This revealed to be useful in Political Science to study, in formal models of political economy, what can be the outcome of a democratic process.