A model for global compactness

Abstract

In a classical paper by Ben-David and Magidor, a model of set theory was exhibited in which ${\aleph }_{\ddot{I}‰+1}$ carries a uniform ultrafilter that is $\hat{I}¸$-indecomposable for every uncountable cardinal $\hat{I}¸<{\aleph }_{\ddot{I}}‰$. 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 $\hat{I}»$, there exists a uniform ultrafilter on $\hat{I}{»}^{+}$ that is $\hat{I}¸$-indecomposable for every cardinal $\hat{I}¸$ such that $cf(\hat{I}»)<\hat{I}¸<\hat{I}»$. 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.

Final version