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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
##### Keywords:
$$n$$-existentially closed
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##### References:
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