On the Very Weak 0-1 Law for Random Graphs with Orders

Let us draw a graph R on {0, 1, . . . , n− 1} by having an edge {i, j} with probability p|i−j|, where ∑ i pi < ∞, and let Mn = (n,<,R). For a first order sentence ψ let aψ be the probability of Mn |= ψ. We know that the sequence aψ, a 2 ψ, . . . , a n ψ, . . . does not necessarily converge. But here we find a weaker substitute which we call the very weak 0-1 law. We prove that limn→∞(a n ψ − a n+1 ψ ) = 0. For this we need a theorem on the (first order) theory of distorted sum of models. Saharon: 1) Check: line before rest of the proof 3.4A, should it be 6=? 2) Check the numerical bounds. Research partially supported by the Binational Science Foundation and partially sponsored by the Edmund Landau Center for research in Mathematical Analysis, supported by the Minerva Foundation (Germany). Done fall 91, Publication 463.