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On an adjacency property of almost all tournaments. (English) Zbl 1108.05048
The authors say a tournament \(T\) is \(k\)-existentially closed (or \(k\)-e.c.) if for every ordered pair of disjoint subsets \(A\) and \(B\) of the nodes of \(T\) with \(|A\cup B|= k\), there exists at least one node \(q\) that beats all nodes of \(A\) and loses to all nodes of \(B\). It follows from results of P. Erdős and L. Moser [Can. Math. Bull. 7, 351–356 (1964; Zbl 0129.34701)] almost all (large) tournaments \(T\) are \(k\)-e.c. for any fixed integer \(k\). The authors of the present paper show that there exists a 2-e.c. tournament \(T\) with \(n\) nodes if and only if \(n= 7\) or \(n\geq 9\).

MSC:
05C20 Directed graphs (digraphs), tournaments
05C35 Extremal problems in graph theory
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