A non-implication between fragments of Martin’s Axiom related to a property which comes from Aronszajn trees

Uniformizing ladder system colorings and the rectangle refining property

“…A typical forcing notion with this property is a(T ), which is the forcing notion adding an antichain through an Aronszajn tree T by finite approximations, and we note that MA ℵ 1 restricted to forcing notions with the property R 1,ℵ 1 implies that every Aronszajn tree is special. The author proved that it is consistent that every forcing notion with the property R 1,ℵ 1 has precaliber ℵ 1 and there exists a non-special Aronszajn tree; hence MA ℵ 1 restricted to forcing notions with the property R 1,ℵ 1 fails [15,Theorem 6.2]. This is quite different from Todorčević and Veličković's theorem [11,Theorem 3.4].…”

“…In this paper, as in [15], forcing notions mean partial orders, and we focus on forcing notions of size ℵ 1 because we will study fragments of MA ℵ 1 , Martin's Axiom for ℵ 1 many dense sets. We consider the following types of forcing notions, which are slightly different from FSCO in [15] (see below).…”

“…In [15], the author introduced a new chain condition, called the property R 1,ℵ 1 , which is stronger than the countable chain condition. A typical forcing notion with this property is a(T ), which is the forcing notion adding an antichain through an Aronszajn tree T by finite approximations, and we note that MA ℵ 1 restricted to forcing notions with the property R 1,ℵ 1 implies that every Aronszajn tree is special.…”

“…The rectangle refining property is introduced by Larson and Todorčević in [7] for partitions on [ω 1 ] 2 , and they introduced the axiom K 2 (rec) which is the statement that every partition with the rectangle refining property has an uncountable homogeneous set. In [12], the author introduced a property for forcing notions (called the anti-rectangle refining property) which is closely related to the rectangle refining property for partitions, and in [15], he introduced the rectangle refining property for forcing notions. In [15], there are some examples.…”

“…In [12], the author introduced a property for forcing notions (called the anti-rectangle refining property) which is closely related to the rectangle refining property for partitions, and in [15], he introduced the rectangle refining property for forcing notions. In [15], there are some examples. A typical forcing notion with this property is also a(T ).…”