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**Zentralblatt MATH**

**Publications of (and about) Paul Erdös**

**Zbl.No: ** 267.05104

**Autor: ** Erdös, Paul; Fried, E.; Hajnal, András; Milner, E.C.

**Title: ** Some remarks on simple tournaments. (In English)

**Source: ** Algebra universalis 2, 238-245 (1972).

**Review: ** A tournament consists of a set T of points on which is defined a complete, anti-symmetric, irreflexive binary relation \rho. A non-empty proper subset X of T is convex if for each y in T-X either x \rho y for all x in X or y \rho x for all x in X. A tournament is simple if it has no convex subsets with more than one point. The authors prove that almost all finite tournaments are simple and that for any tournament T with |T| \ne 2 there exists a simple tournament R such that T \subset R and |R-T| = 2. (Criteria for a tournament to have a simple one-point extension have been given by the reviewer [Discrete Math. 2, 389-395 (1972; Zbl 236.05108)] when T is finite and by *P. Erdös, A. Hajnal* and *E. C. Milner* [Mathematika, London 19, 57-62 (1972; Zbl 242.05113)] in the general case.)

**Reviewer: ** J.W.Moon

**Classif.: ** * 05C20 Directed graphs (digraphs)

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