Hereditary Zero-One Laws for Graphs

Abstract

We consider the random graph $M^n_{\bar{p}}$ on the set [n], where the probability of {x,y} being an edge is p_{|x − y|}, and $\bar{p}=(p_1,p_2,p_3,...)$ is a series of probabilities. We consider the set of all $\bar{q}$ derived from $\bar{p}$ by inserting 0 probabilities into $\bar{p}$, or alternatively by decreasing some of the p _i . We say that $\bar{p}$ hereditarily satisfies the 0-1 law if the 0-1 law (for first order logic) holds in $M^n_{\bar{q}}$ for every $\bar{q}$ derived from $\bar{p}$ in the relevant way described above. We give a necessary and sufficient condition on $\bar{p}$ for it to hereditarily satisfy the 0-1 law.