A model for global compactness

Joint work with Sittinon Jirattikansakul and Inbar Oren.

Abstract. In a classical paper by Ben-David and Magidor, a model of set theory was exhibited in which ${\mathrm{\aleph }}_{\omega +1}$ carries a uniform ultrafilter that is $\theta$-indecomposable for every uncountable $\theta <{\mathrm{\aleph }}_{\omega }$.

In this paper, we give a global version of this result, as follows:

Assuming the consistency of a supercompact cardinal, we produce a model of set theory in which for every singular cardinal $\lambda$, there exists a uniform ultrafilter on ${\lambda }^{+}$ that is $\theta$-indecomposable for every cardinal $\theta$ with $cf(\lambda )<\theta <\lambda$. In our model, many instances of compactness for chromatic numbers hold, from which we infer that Hajnal’s gap-1 counterexample to Hedetniemi’s conjecture is best possible on the grounds of ZFC.

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