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

Let Gn be the binomial random graph G(n,p=c/n) in the sparse regime, which as is well‐known undergoes a phase transition at c=1 . Lynch ( RandomStructuresandAlgorithms , 1992) showed that for every first order sentence ϕ , the limiting probability that Gn satisfies ϕ as n→∞ exists, and moreover it is an analytic function of c. In this paper we consider the closure Lc‾ in the interval [0,1] of the set Lc of all limiting probabilities of first order sentences in Gn . We show that there exists a critical value c0≈0.93 such that Lc‾=[0,1] when c≥c0 , whereas Lc‾ misses at least one subinterval when c

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