对Erdös和moser关于比赛的猜想的反驳

Journal of Combinatorial Theory Pub Date : 1970-10-01 DOI:10.1016/S0021-9800(70)80061-8

K.B. Reid , E.T. Parker

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引用次数: 70

摘要

Erdös和Moser[1]给出了一个没有4阶传递子竞赛的7阶竞赛，并推测对于每一个正整数k存在一个没有k阶传递子竞赛的2k−1−1阶竞赛。对于k=5，该猜想被证明是错误的。而且，每一个14阶的比武都有一个5阶的可传递子比武。归纳地说，这个猜想对所有大于5的阶都是假的。给出了一个不存在5阶可传递子斗笠的13阶斗笠的存在唯一性。

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Disproof of a conjecture of Erdös and moser on tournaments

Erdös and Moser [1] displayed a tournament of order 7 with no transitive subtournament of order 4 and conjectured for each positive integer k existence of a tournament of order 2^k−1−1 with no transitive subtournament of order k. The conjecture is disproved for k=5. Further, every tournament of order 14 has a transitive subtournament of order 5. Inductively, the conjecture is false for all orders above 5. Existence and uniqueness of a tournament of order 13 having no transitive subtournament of order 5 are shown.

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Journal of Combinatorial Theory

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