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Preprint Number 482

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482. Ove Ahlman and Vera Koponen
Random l-colourable structures with a pregeometry

Submission date: 23 July 2012.


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).

Mathematics Subject Classification: 03C13

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Full text arXiv 1207.4936: pdf, ps.

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