On an adjacency property of almost all tournaments

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 SS, there is a vertex x∉Sx∉S which is directed toward every vertex in T   and directed away from every vertex in S⧹TS⧹T. We prove that there is a 2-e.c. tournament with k   vertices if and only if k≥7k≥7 and k≠8k≠8, and give explicit examples for all such orders k. We also give a replication operation which preserves the 2-e.c. property.