## 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 a^{n}[psi] be the probability of Mn [models] [psi]. We know that the sequence a^{1}[psi],a^{2}[psi],...,a^{n}[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](a^{n}[psi] - a^{n+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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