Random l-colourable structures with a pregeometry

TL;DR

This paper investigates the properties of finite l-colourable structures with pregeometry, establishing a zero-one law, explicit axiomatization, and definability of colour relations under a specific probabilistic model.

Contribution

It introduces a probabilistic model for generating l-colourable structures with pregeometry and proves key logical properties including a zero-one law and definability results.

Findings

Proves a zero-one law for the class of structures.

Provides an explicit axiomatization for almost sure properties.

Shows the colour relation is definable by a specific formula.

Abstract

We study finite $l$-colourable structures with an underlying pregeometry. The probability measure that is used corresponds to a process of generating such structures (with a given underlying pregeometry) by which colours are first randomly assigned to all 1-dimensional subspaces and then relationships are assigned in such a way that the colouring conditions are satisfied but apart from this in a random way. We can then ask what the probability is that the resulting structure, where we now forget the specific colouring of the generating process, has a given property. With this measure we get the following results: 1. A zero-one law. 2. The set of sentences with asymptotic probability 1 has an explicit axiomatisation which is presented. 3. There is a formula $\xi(x,y)$ (not directly speaking about colours) such that, with asymptotic probability 1, the relation "there is an $l$-colouring which assigns the same colour to $x$ and $y$" is defined by $\xi(x,y)$. 4. With asymptotic probability 1, an $l$-colourable structure has a unique $l$-colouring (up to permutation of the colours).