Random ℓ‐colourable structures with a pregeometry

We study finite ℓ‐colourable structures with an underlying pregeometry. The probability measure that is used corresponds to a process of generating such structures 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).