Archive for Mathematical Logic

, Volume 56, Issue 7–8, pp 783–796 | Cite as

Same graph, different universe

Article

Abstract

May the same graph admit two different chromatic numbers in two different universes? How about infinitely many different values? and can this be achieved without changing the cardinals structure? In this paper, it is proved that in Gödel’s constructible universe, for every uncountable cardinal \(\mu \) below the first fixed-point of the \(\aleph \)-function, there exists a graph \(\mathcal G_\mu \) satisfying the following:
  • \(\mathcal G_\mu \) has size and chromatic number \(\mu \);

  • for every infinite cardinal \(\kappa <\mu \), there exists a cofinality-preserving \({{\mathrm{GCH}}}\)-preserving forcing extension in which \({{\mathrm{Chr}}}(\mathcal G_\mu )=\kappa \).

Keywords

C-sequence graph Chromatic spectrum Mutual stationarity Cardinal fixed-point 

Mathematics Subject Classification

Primary 03E35 Secondary 05C15 05C63 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Department of MathematicsBar-Ilan UniversityRamat GanIsrael

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