Sufficient Conditions for Labelled 0-1 Laws

Abstract

If F(x) = e G(x), where F(x) = Σf(n)x n and G(x) = Σg(n)x n, with 0≤g(n) = O(n θn /n!), θ∈(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 [Compton 1987] 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 θn) for some θ∈(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.