Hereditary zero-one laws for graphs

Abstract

We consider the random graph on the set [n], where the probability of {x,y} being an edge is p |x-y|, and is a series of probabilities. We consider the set of all derived from by inserting 0 probabilities into, or alternatively by decreasing some of the p i . We say that hereditarily satisfies the 0-1 law if the 0-1 law (for first order logic) holds in Mnq̄ for every derived from in the relevant way described above. We give a necessary and sufficient condition on for it to hereditarily satisfy the 0-1 law. © 2010 Springer-Verlag Berlin Heidelberg.