Martin's Axiom and Weak Kurepa Hypothesis

TL;DR

This paper demonstrates that it is consistent for Martin's Axiom to hold at 2 while the weak Kurepa Hypothesis fails, using a model constructed from a Mahlo cardinal.

Contribution

It provides a consistency proof showing Martin's Axiom at 2 can coexist with the failure of the weak Kurepa Hypothesis, answering an open question.

Findings

Martin's Axiom at 2 can hold while the weak Kurepa Hypothesis fails.

The model constructed ensures the failure of the weak Kurepa Hypothesis in any c.c.c. forcing extension.

The result is relative to the existence of a Mahlo cardinal.

Abstract

I show that it is consistent relative to the consistency of a Mahlo cardinal that Martin's axiom holds at $\omega_2$, but the weak Kurepa Hypothesis fails. This answers a question posed by Honzik, Lambie-Hanson and Stejskalov\'a. The consistency result is obtained by constructing a model where the weak Kurepa Hypothesis fails in any c.c.c. forcing extension.