How Complex are Random Graphs in First Order Logic?
Combinatorics
2007-05-23 v1
Abstract
It is not hard to write a first order formula which is true for a given graph G but is false for any graph not isomorphic to G. The smallest number $(G) of nested quantifiers in a such formula can serve as a measure for the ``first order complexity'' of G. Here, this parameter is studied for random graphs. We determine it asymptotically when the edge probability p is constant; in fact, D(G) is of order log n then. For very sparse graphs its magnitude is \Theta(n). On the other hand, for certain (carefully chosen) values of p the parameter D(G) can drop down to the very slow growing function log^* n, the inverse of the tower-function. The general picture, however, is still a mystery.
Keywords
Cite
@article{arxiv.math/0401247,
title = {How Complex are Random Graphs in First Order Logic?},
author = {Jeong Han Kim and Oleg Pikhurko and Joel Spencer and Oleg Verbitsky},
journal= {arXiv preprint arXiv:math/0401247},
year = {2007}
}
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27 pages