Asymptotic probabilities of extension properties and random l-colourable structures

We consider a setK = ⋃ n∈N Kn of nite structures such that all members of Kn have the same universe, the cardinality of which approaches ∞ as n → ∞. Each structure in K may have a nontrivial underlying pregeometry and on each Kn we consider a probability measure, either the uniform measure, or what we call the dimension conditional measure. The main questions are: What conditions imply that for every extension axiom φ, compatible with the de ning properties of K, the probability that φ is true in a member of Kn approaches 1 as n → ∞? And what conditions imply that this is not the case, possibly in the strong sense that the mentioned probability approaches 0 for some φ? If each Kn is the set of structures with universe {1, . . . , n}, in a xed relational language, in which certain forbidden structures cannot be weakly embedded and K has the disjoint amalgamation property, then there is a condition (concerning the set of forbidden structures) which, if we consider the uniform measure, gives a dichotomy; i.e. the condition holds if and only if the answer to the rst question is `yes'. In general, we do not obtain a dichotomy, but we do obtain a condition guaranteeing that the answer is `yes' for the rst question, as well as a condition guaranteeing that the answer is `no'; and we give examples showing that in the gap between these conditions the answer may be either `yes' or `no'. This analysis is made for both the uniform measure and for the dimension conditional measure. The later measure has closer relation to random generation of structures and is more generous with respect to satis ability of extension axioms. Random l-coloured structures fall naturally into the framework discussed so far, but random l-colourable structures need further considerations. It is not the case that every extension axiom compatible with the class of l-colourable structures almost surely holds in an l-colourable structure. But a more restricted set of extension axioms turns out to hold almost surely, which allows us to prove a zero-one law for random l-colourable structures, using a probability measure which is derived from the dimension conditional measure, and, after further combinatorial considerations, also for the uniform probability measure.