Fragments of Martin's axiom

TL;DR

This paper explores the equivalence between Martin's axiom for ense sets and a fragment involving the Knaster property K, demonstrating the minimality of the dimension 3 in K.

Contribution

It establishes the equivalence of Martin's axiom for ense sets with a fragment involving the Knaster property K and shows the minimality of the dimension 3 in K.

Findings

Martin's axiom for ense sets is equivalent to a K property for ccc posets.

The dimension 3 in K is shown to be minimal.

The paper clarifies the relationship between Martin's axiom and the Knaster property.

Abstract

We show that Martin's axiom for $\omega_1$ dense sets is equivalent to its fragment asserting that every ccc poset has the Knaster property K$_3$. On the other hand, we show that the dimension 3 in K$_3$ is in some sense minimal.