A model for global compactness

Abstract

In a classical paper by Ben-David and Magidor, a model of set theory was exhibited in which $\aleph_{\omega+1}$ carries a uniform ultrafilter that is $\theta$-indecomposable for every uncountable cardinal $\theta<\aleph_\omega$. 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 $\theta$ such that $cf(\lambda)<\theta<\lambda$. 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.