How complex are random graphs in first order logic?

Jeong Han Kim, Oleg Pikhurko, Joel H. Spencer, Oleg Verbitsky

Research output: Contribution to journalArticle

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 D(G) of nested quantifiers in such a 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.

Original languageEnglish (US)
Pages (from-to)119-145
Number of pages27
JournalRandom Structures and Algorithms
Volume26
Issue number1-2
DOIs
StatePublished - Jan 2005

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First-order Logic
Random Graphs
First-order
Sparse Graphs
Quantifiers
Graph in graph theory
Isomorphic
False

ASJC Scopus subject areas

  • Computer Graphics and Computer-Aided Design
  • Software
  • Mathematics(all)
  • Applied Mathematics

Cite this

How complex are random graphs in first order logic? / Kim, Jeong Han; Pikhurko, Oleg; Spencer, Joel H.; Verbitsky, Oleg.

In: Random Structures and Algorithms, Vol. 26, No. 1-2, 01.2005, p. 119-145.

Research output: Contribution to journalArticle

Kim, Jeong Han ; Pikhurko, Oleg ; Spencer, Joel H. ; Verbitsky, Oleg. / How complex are random graphs in first order logic?. In: Random Structures and Algorithms. 2005 ; Vol. 26, No. 1-2. pp. 119-145.
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