Asymptotic probabilities of extension properties and random $l$-colourable structures

TL;DR

This paper investigates the asymptotic probabilities of extension properties in random finite structures with pregeometry, establishing conditions under which these probabilities tend to 1 or 0, with implications for random $l$-colourable structures.

Contribution

It provides a comprehensive analysis of when extension axioms hold with high probability in random structures, introducing conditions for dichotomy and non-dichotomy scenarios under different measures.

Findings

A dichotomy condition for uniform measure in structures with forbidden substructures.

Conditions guaranteeing high probability of extension axioms.

Examples illustrating the gap where probabilities may tend to 0 or 1.

Abstract

We consider a set $\mbK = \bigcup_{n \in \mbbN}\mbK_n$ of {\em finite} structures such that all members of $\mbK_n$ have the same universe, the cardinality of which approaches $\infty$ as $n\to\infty$. Each structure in $\mbK$ may have a nontrivial underlying pregeometry and on each $\mbK_n$ we consider a probability measure, either the uniform measure, or what we call the {\em dimension conditional measure}. The main questions are: What conditions imply that for every extension axiom $\varphi$, compatible with the defining properties of $\mbK$, the probability that $\varphi$ is true in a member of $\mbK_n$ approaches 1 as $n \to \infty$? And what conditions imply that this is not the case, possibly in the strong sense that the mentioned probability approaches 0 for some $\varphi$? If each $\mbK_n$ is the set of structures with universe ${1, ..., n}$, in a fixed relational language, in which certain "forbidden" structures cannot be weakly embedded and $\mbK$ has the disjoint amalgamation property, then there is a condition (concerning the set of forbidden structures) which, if we consider the uniform measure, gives a dichotomy; i.e. the condition holds if and only if the answer to the first question is `yes'. In general, we do not obtain a dichotomy, but we do obtain a condition guaranteeing that the answer is `yes' for the first question, as well as a condition guaranteeing that the answer is `no'; and we give examples showing that in the gap between these conditions the answer may be either `yes' or `no'. This analysis is made for both the uniform measure and for the dimension conditional measure. The later measure has closer relation to random generation of structures and is more "generous" with respect to satisfiability of extension axioms.