Cicho\'n's maximum

TL;DR

This paper demonstrates, assuming four strongly compact cardinals, that it is consistent for all entries in Cichoń's diagram to be pairwise distinct, establishing a maximal separation of these cardinal characteristics.

Contribution

It proves the consistency of a maximal separation of Cichoń's diagram entries under large cardinal assumptions, extending previous results in set theory.

Findings

All entries in Cichoń's diagram can be pairwise different.

The separation is achieved assuming four strongly compact cardinals.

The resulting chain of inequalities is consistent with ZFC.

Abstract

Assuming four strongly compact cardinals, it is consistent that all entries in Cicho\'n's diagram are pairwise different, more specifically that \[ \aleph_1 < \mathrm{add}(\mathrm{null}) < \mathrm{cov}(\mathrm{null}) < \mathfrak{b} < \mathrm{non}(\mathrm{meager}) < \mathrm{cov}(\mathrm{meager}) < \mathfrak{d} < \mathrm{non}(\mathrm{null}) < \mathrm{cof}(\mathrm{null}) < 2^{\aleph_0}.\]