A model for global compactness

TL;DR

This paper constructs a set-theoretic model assuming supercompact cardinals where ultrafilters exhibit strong indecomposability properties across all singular cardinals, impacting combinatorial graph theory and conjectures.

Contribution

It provides a global extension of a classical set theory model, demonstrating ultrafilter properties for all singular cardinals under large cardinal assumptions.

Findings

Existence of ultrafilters with uniform indecomposability on successor cardinals of singulars.

Many instances of compactness for chromatic numbers are established.

Hajnal's gap-1 counterexample to Hedetniemi's conjecture is shown to be optimal in ZFC.

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.