First order zero-one laws for random graphs on the circle

Abstract

We look at a competitor of the Erdős–Renyi models of random graphs, one proposed in E. Gilbert [J. Soc. Indust. Appl. Math. 9:4, 533–543 (1961)]: given δ>0 and a metric space X of diameter >δ, scatter n vertices at random on X and connect those of distance <δ apart: we get a random graph G. Letting X be a circle, we look at zero-one laws for (in First Order Logic) various δ. ©1999 John Wiley & Sons, Inc. Random Struct. Alg., 14, 239–266, 1999