Two chain conditions and their Todorčević's fragments of Martin's Axiom

Abstract

In this paper, we investigate two chain conditions of forcing notions, called the rectangle refining property and property R1,ℵ1. They are stronger than the countable chain condition. Some of their typical examples are forcing notions about indestructible gaps introduced by Kunen. Both chain conditions are similar and have common examples, however, no distinction between them is known so far.In this paper, we propose a difference of two chain conditions from the view point of Todorčević's fragments of Martin's Axiom. More precisely, it is proved that some Todorčević's fragment of Martin's Axiom for forcing notions with property R1,ℵ1 is consistent with the existence of a non-special Aronszajn tree, and that some Todorčević's fragment of Martin's Axiom for forcing notions with the rectangle refining property in some weak sense implies that every Aronszajn tree is special.