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arXiv · 1207.4936

Random l-colourable structures with a pregeometry

Ove Ahlman·Vera Koponen
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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 $ξ(x,y)$ (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 $ξ(x,y)$. 4. With asymptotic probability 1, an $l$-colourable structure has a unique $l$-colouring (up to permutation of the colours).

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Ove Ahlman, Vera Koponen. 2012-07-20. Random l-colourable structures with a pregeometry. https://doi.org/10.1002/malq.201500006

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