Very weak zero one law for random graphs with order and random binary functions

by Shelah. [Sh:548]
Random Structures \& Algorithms, 1996
Let G_<(n,p) denote the usual random graph G(n,p) on a totally ordered set of n vertices. We will fix p= frac {1}{2} for definiteness. Let L^< denote the first order language with predicates equality (x=y), adjacency (x~y) and less than (x<y) . For any sentence A in L^< let f_A(n) denote the probability that the random G_<(n,p) has property A . It is known Compton, Henson and Shelah [CHSh:245] that there are A for which f_A(n) does not converge. Here we show what is called a very weak zero-one law (from [Sh 463]): THEOREM: For every A in language L^<, lim_{n-> infty}(f_A(n+1)-f_A(n))=0.

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