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The number of irreducible tournaments

  • E. M. Wright (a1)

Extract

An n-tournament is a set of n labelled points, each pair A, B of which is joined either by the oriented line AB or by the oriented line BA. There are N = n(n –1)/2 such pairs and so Fn different n-tournaments, where Fn = 2N. A tournament is reducible if the points can be separated into two non-empty subsets and ℬ, such that every line joining a point in to a point in ℬ is directed towards the point in ℬ. Rado [3] showed that an irreducible tournamentis strongly connected; i.e. for every ordered pair of points A, B, there is a sequence of correctly oriented lines AC1, C1C2, …, ChB in the tournament, and conversely that a strongly connected tournament is irreducible.

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References

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1.Moon, J. W. and Moser, L., Almost all tournaments are irreducible, Canadian Math. Bull. 5 (1962), 6165.
2.Moon, J. W., Topics on tournaments (New York, 1968).
3.Rado, R., Theorems on linear combinatorial topology and general measure, Ann. of Math. 44 (1943), 228270.

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