Controlling classical cardinal characteristics while collapsing cardinals

Abstract

We show how to force distinct values to m, p and h and the values in Cichoń’s diagram, using the Boolean Ultrapower method. In our recent paper [J. Math. Logic 21 (2021)] the same was done for a newer Cichoń’s Maximum construction which does not require large cardinals. The present version does need large cardinals, but allows one more value, in addition to the continuum, to be singular (either cov(M) or d). We also show the following: Given a forcing notion P that forces certain values to several classical cardinal characteristics of the reals, we can compose P with a collapse (of a cardinal λ > κ to κ) such that the composition still forces the previous values to these characteristics.