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Hereditary Zero-One Laws for Graphs

  • Saharon Shelah
  • Mor Doron
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6300)

Abstract

We consider the random graph \(M^n_{\bar{p}}\) on the set [n], where the probability of {x,y} being an edge is p|x − y|, and \(\bar{p}=(p_1,p_2,p_3,...)\) is a series of probabilities. We consider the set of all \(\bar{q}\) derived from \(\bar{p}\) by inserting 0 probabilities into \(\bar{p}\), or alternatively by decreasing some of the p i . We say that \(\bar{p}\) hereditarily satisfies the 0-1 law if the 0-1 law (for first order logic) holds in \(M^n_{\bar{q}}\) for every \(\bar{q}\) derived from \(\bar{p}\) in the relevant way described above. We give a necessary and sufficient condition on \(\bar{p}\) for it to hereditarily satisfy the 0-1 law.

Keywords

random graphs zero-one laws 

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References

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    Łuczak, T., Shelah, S.: Convergence in homogeneous random graphs. Random Structures Algorithms 6(4), 371–391 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
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    Shelah, S.: Hereditary convergence laws with successor (in preparation)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Saharon Shelah
    • 1
  • Mor Doron
    • 1
  1. 1.Department of MathematicsThe Hebrew University of JerusalemIsrael

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