On an adjacency property of almost all tournaments Original Research Article

Let n be a positive integer. A tournament is called n-existentially closed (or n-e.c.) if for every subset S of n vertices and for every subset T of image, there is a vertex image which is directed toward every vertex in T and directed away from every vertex in image. We prove that there is a 2-e.c. tournament with k vertices if and only if image and image, and give explicit examples for all such orders k. We also give a replication operation which preserves the 2-e.c. property.