Distinguishing Martin's axiom from its restrictions

TL;DR

This paper introduces a new forcing iteration method to differentiate Martin's axiom from its restrictions, showing that the full axiom is strictly stronger than its restrictions to certain classes of forcing notions.

Contribution

The paper presents a novel forcing iteration technique that demonstrates Martin's axiom is strictly stronger than its restrictions to ccc forcing notions and their squares.

Findings

Martin's axiom is strictly stronger than its restriction to ccc forcing notions in all finite powers.

Martin's axiom is strictly stronger than its restriction to forcing notions with ccc squares.

The new method minimally damages strong colorings during forcing iterations.

Abstract

We introduce an iteration of forcing notions satisfying the countable chain condition with minimal damage to a strong coloring. Applying this method, we prove that Martin's axiom is strictly stronger than its restriction to forcing notions satisfying the countable chain condition in all finite powers. Our method shows also the finer distinction, that Martin's axiom is strictly stronger than its restriction to forcing notions whose squares satisfy the countable chain condition.