Partition Calculus and Cardinal Invariants

Shimon Garti, Saharon Shelah

Published 2011-12-25, updated 2012-11-27Version 3

We prove that the strong polarized relation of $\theta$ above $\omega$ applied simultaneously for every cardinal in the interval $[\aleph_1,\aleph]$ is consistent. We conclude that this positive relation is consistent for every cardinal invariant on the continuum. We show that similar results hold for a supercompact cardinal, and for a strong limit singular under some assumptions.