Sufficient Conditions for Labelled 0-1 Laws

Abstract

If F(x) = e^G(x), where F(x) = \sum f(n)x^n and $G(x) = \sum g(n)x^n, with 0 \le g(n) = O(n^{theta n}/n!),theta in (0,1), and gcd(n : g(n) > 0)=1, then f(n) = o(f(n-1)). This gives an answer to Compton's request in Question 8.3 for an ``easily verifiable sufficient condition'' to show that an adequate class of structures has a labelled first-order 0-1 law, namely it suffices to show that the labelled component count function is O(n^{theta n}) for some theta in (0,1). It also provides the means to recursively construct an adequate class of structures with a labelled 0-1 law but not an unlabelled 0-1 law, answering Compton's Question 8.4.