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Infinitary logics and very sparse random graphs

Published online by Cambridge University Press:  12 March 2014

James F. Lynch
James F. Lynch*
Affiliation:
Department of Mathematics and Computer Science, Clarkson University, Potsdam, NY 13699-5815, USA, E-mail: jlynch@sun.mcs.clarkson.edu
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Abstract

Let be the infinitary language obtained from the first-order language of graphs by closure under conjunctions and disjunctions of arbitrary sets of formulas, provided only finitely many distinct variables occur among the formulas. Let p(n) be the edge probability of the random graph on n vertices. It is shown that if p(n) ≪ n−1 satisfies certain simple conditions on its growth rate, then for every , the probability that σ holds for the random graph on n vertices converges. In fact, if p(n) = n−α, α > 1, then the probability is either smaller than for some d > 0, or it is asymptotic to cn−d for some c > 0, d ≥ 0. Results on the difficulty of computing the asymptotic probability are given.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 62 , Issue 2 , June 1997 , pp. 609 - 623
Copyright
Copyright © Association for Symbolic Logic 1997

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