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How Complex are Random Graphs in First Order Logic?

Jeong Han Kim·Oleg Pikhurko·Joel Spencer·Oleg Verbitsky
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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 Θ(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.

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Jeong Han Kim, Oleg Pikhurko, Joel Spencer, Oleg Verbitsky. 2004-01-20. How Complex are Random Graphs in First Order Logic?. https://arxiv.org/abs/math/0401247

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