Hereditary Zero-One Laws for Graphs

TL;DR

This paper studies hereditary zero-one laws in a class of random graphs defined by edge probabilities depending on vertex distance, providing a necessary and sufficient condition for these laws to hold across derived probability series.

Contribution

It introduces a new hereditary property for zero-one laws in random graphs and characterizes exactly when this property holds based on the probability series.

Findings

Provides a necessary and sufficient condition for hereditary zero-one laws.

Defines a class of random graphs with distance-dependent edge probabilities.

Establishes the stability of zero-one laws under probability modifications.

Abstract

We consider the random graph M^n_{\bar{p}} on the set [n], were 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 to \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 any \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.