Forcing constellations of Cicho\'n's diagram by using the Tukey order

TL;DR

This paper employs finite support iteration and Tukey order techniques to construct models with diverse configurations of cardinal characteristics in Cichoń's diagram, demonstrating how to force specific values and maximums within the diagram.

Contribution

It introduces new methods for forcing various constellations of Cichoń's diagram using the Tukey order and restriction techniques to control cardinal characteristics.

Findings

Constructed models with pairwise different cardinal characteristics.

Forced the left-hand side of Cichoń's diagram.

Presented techniques to achieve the maximum of Cichoń's diagram.

Abstract

We use known finite support iteration techniques to present various examples of models where several cardinal characteristics of Cicho\'n's diagram are pairwise different. We show some simple examples forcing the left-hand side of Cicho\'n's diagram, and present the technique of restriction to models to force Cicho\'n's maximum (original from Goldstern, Kellner, Shelah, and the second author). We focus on how the values forced in all the constellations are obtained via the Tukey order.