The Maximum Vertex Degree of a Graph on Uniform Points in [0, 1] d

On independent random points U 1 ,· ··,U n distributed uniformly on [0, 1] d , a random graph G n ( x ) is constructed in which two distinct such points are joined by an edge if the l ∞ -distance between them is at most some prescribed value 0 ≦ x ≦ 1. Almost-sure asymptotic rates of convergence/divergence are obtained for the maximum vertex degree of the random graph and related quantities, including the clique number, chromatic number and independence number, as the number n of points becomes large and the edge distance x is allowed to vary with n. Series and sequence criteria on edge distances { x n } are provided which guarantee the random graph to be empty of edges, a.s.