Counting finite models

Let phi be a monadic second order sentence about a finite structure from a class K which is closed under disjoint unions and has components. Compton has conjectured that if the number of it element structures has appropriate asymptotics, then unlabelled (labelled) asymptotic probabilities v(phi) (mu(phi) respectively) for phi always exist. By applying generating series methods to count finite models, and a tailor made Tauberian lemma, this conjecture is proved under a mild additional condition on the asymptotics of the number of single component K-structures. Prominent among examples covered, are structures consisting of a single unary function (or partial function) and a fixed number of unary predicates.