Volume 6: January - December 1996

Issue 1: 1996

Abstract


On the very weak 0-1 law for random graphs with orders

  • On the very weak 0-1 law for random graphs with orders
  • S. Shelah The Hebrew University, Mathematics Institute, Israel and Mathematics Department, Rutgers University, USA

    ABSTRACT

    Let us draw a graph R on {0,1,...n-1} by having an edge {i,j} with probability p|i-j|, where [Sigma]ipi < [infinity], and let Mn = (n,<,R). For a first-order sentence [psi] let an[psi] be the probability of Mn [models] [psi]. We know that the sequence a1[psi],a2[psi],...,an[psi],...does not necessarily converge. But here we find a weaker substitute which we call the very weak 0-1 law. We prove that limn -> [infinity](an[psi] - an+1[psi]) = 0. For this we need a theorem on the (first-order) theory of distorted sum of models.

    Pages: 139 - 161

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    Copyright Oxford University Press, 1996