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:
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\(\mathcal G_\mu \) has size and chromatic number \(\mu \);
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for every infinite cardinal \(\kappa <\mu \), there exists a cofinality-preserving \({{\mathrm{GCH}}}\)-preserving forcing extension in which \({{\mathrm{Chr}}}(\mathcal G_\mu )=\kappa \).
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Rinot, A. Same graph, different universe. Arch. Math. Logic 56, 783–796 (2017). https://doi.org/10.1007/s00153-017-0551-x
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DOI: https://doi.org/10.1007/s00153-017-0551-x