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

by Shelah. [Sh:463]
J Logic and Computation, 1996
Let us draw a graph R on {0,1,...,n-1} by having an edge {i,j} with probability p_(|i-j|), where sum_i p_i is finite and let M_n=(n,<,R) . For a first order sentence psi let a^n_psi be the probability of ``M_n satisfies psi''. We prove that the limit of a^n_psi-a^{n+1}_psi is 0, as n goes to infinity.


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