On 2-e.c. graphs, tournaments, and hypergraphs

We study adjacency properties in the class of k-colourable graphs, the class of Km-free graphs, and the class of tournaments. In particular, we investigate the 2-existentially closed (or 2-e.c.) adjacency property in these classes which was studied in the class of graphs in [6], [7] and [8]. For a fixed integer n ⩾ 1, a graph G is called n-existentially closed or n-e.c. if for for every n-element subset S of the vertices, and for every subset T of S, there is a vertex not in S which is joined to every vertex in T and to no vertex in S \ T. In each of the classes described above, we determine the possible orders of the 2-e.c. members of the class, and give explicit examples for all such orders. Some results on adjacency properties in the class of k-uniform hypergraphs are discussed.