Burris, Stanley N. and Yeats, Karen A. - Sufficient conditions for labelled 0-1 laws

dmtcs:430 - Discrete Mathematics & Theoretical Computer Science, January 1, 2008, Vol. 10 no. 1
Sufficient conditions for labelled 0-1 laws

Authors: Burris, Stanley N. and Yeats, Karen A.

If F(x) = e^G(x), where F(x) = \Sum f(n)x^n and G(x) = \Sum g(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.


Source : oai:HAL:hal-00972301v1
Volume: Vol. 10 no. 1
Section: Combinatorics
Published on: January 1, 2008
Submitted on: March 26, 2015
Keywords: [INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]


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