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(infinity)-distance between them is at most some prescribed value 0 less than or equal to x less than or equal to 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.