Abstract

Let $\varphi$ be a monadic second order sentence about a finite structure from a class $\mathscr{K}$ which is closed under disjoint unions and has components. Compton has conjectured that if the number of $n$ element structures has appropriate asymptotics, then unlabelled asymptotic probabilities $\nu $ respectively) for $\varphi$ 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 $\mathscr{K}$-structures. Prominent among examples covered, are structures consisting of a single unary function and a fixed number of unary predicates.