Limiting probabilities of first order properties of random sparse graphs and hypergraphs

Abstract

AbstractLet be the binomial random graph in the sparse regime, which as is well‐known undergoes a phase transition at . Lynch (, 1992) showed that for every first order sentence , the limiting probability that satisfies as exists, and moreover it is an analytic function of c. In this paper we consider the closure in the interval of the set of all limiting probabilities of first order sentences in . We show that there exists a critical value such that when , whereas misses at least one subinterval when . We extend these results to random sparse d‐uniform hypergraphs, where the probability of a d‐edge is .