Random ?-colourable structures with a pregeometry

摘要

We study finite ?-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 ?-colouring which assigns the same colour to x and y" is defined by ξ (x, y). (4) With asymptotic probability 1, an ?-colourable structure has a unique ?-colouring (up to permutation of the colours).