The vapnik-chervonenkis dimension of a random graph

AbstractIn this paper we investigate a parameter defined for any graph, known as the Vapnik Chervonenkis dimension (VC dimension). For any vertex x of a graph G, the closed neighborhood N(x) of x is the set of all vertices of G adjacent to x, together with x. We say that a set D of vertices of G is shattered if every subset R of D can be realised as R=D∩N(x) for some vertex x of G. The VC dimension of G is defined to be the largest cardinality of a shattered set of vertices. Our main result gives, for each positive integer d, the exact threshold function for a random graph G(n, p) to have VC dimension d