Some weak fragments of Martin’s axiom related to the rectangle refining property

Abstract

We introduce the anti-rectangle refining property for forcing notions and investigate fragments of Martin's axiom for ℵ 1 dense sets related to the anti-rectangle refining property, which is close to some fragment of Martin's axiom for ℵ 1 dense sets related to the rectangle refining property, and prove that they are really weaker fragments. Definition 2.1. • (Knaster, [8]) A forcing notion P has property K if for every I ∈ [P] ℵ1 , there is a refinement I ∈ [I] ℵ1 such that p and q are compatible in P (i.e. there exists r ∈ P such that both r ≤ P p and r ≤ P q hold, denoted by p ⊥ P q) for every p and q in I. • A forcing notion P has the anti-rectangle refining property if it is uncountable and for every I and J in [P] ℵ1 , there are refinements I ∈ [I] ℵ1 and J ∈ [J] ℵ1 such that p and q are incompatible in P (denoted by p ⊥ P q) for every p ∈ I and q ∈ J. For a forcing notion P, we define another forcing notion a(P) which consists of finite antichains in P, ordered by reverse inclusion. If a(P) has the ccc, then by the genericity, a(P) can add an uncountable antichain in P. So it follows from the next lemma that if P has the ccc and the anti-rectangle refining property, then a(P) can destroy the ccc property of P. Proposition 2.2. If a forcing notion P has the anti-rectangle refining property, then for every I and J in the set [a(P)] ℵ1 with I ∪ J pairwise disjoint, i.e. σ ∩ τ = ∅ for any distinct σ and τ in I ∪ J, there are refinements I ∈ [I] ℵ1 and J ∈ [J] ℵ1 such that for any σ ∈ I and τ ∈ J , σ ⊥ a(P) τ , i.e. σ ∪ τ ∈ a(P). This property is very similar to the rectangle refining property. Definition 2.3 (Larson-Todorčević, [11, Definition 4.1.]). A partition [ω 1 ] 2 = K 0 ∪ K 1 satisfies the rectangle refining property if for every I and J in the set [ω 1 ] ℵ1 , there are refinements I ∈ [I] ℵ1 and J ∈ [J] ℵ1 such that {{α, β} ; α ∈ I & β ∈ J & α < β} ⊆ K 0. This is the reason why I call the anti-rectangle refining property. We note that if a partition [ω 1 ] 2 = K 0 ∪ K 1 satisfies the rectangle refining property, then for every I and J in [ω 1 ] ℵ1 , there are refinements I ∈ [I] ℵ1 and J ∈ [J] ℵ1 such that (I and J are disjoint and) {{α, β} ; α ∈ I & β ∈ J } ⊆ K 0. Moreover we should note that this property is stronger than the ccc, so if P has the anti-rectangle refining property, then a(P) has the ccc. Proof of Proposition 2.2. Assume that P has the anti-rectangle refining property and let I and J be uncountable subsets of a(P) such that I ∪ J is pairwise disjoint. By shrinking I and J respectively if need, we may assume that each element of I has size m and each element of J has size n. Applying the anti-rectangle refining property mn times, we can find uncountable subsets I and J of I and J respectively such that for every σ ∈ I and τ ∈ J , the i-th condition in σ and the j-th condition

Rudin's Dowker space in the extension with a Suslin tree

“…It follows from the above example that MA ℵ 1 (rec) implies K 2 (rec). By the same argument in [23], we can show that it is consistent that MA ℵ 1 (rec) holds but MA ℵ 1 fails, so MA ℵ 1 (rec) is strictly weaker than MA ℵ 1 .…”

“…It is a deep question whether MA ℵ 1 (rec) is strictly stronger than K 2 (rec), similarly to other problems on fragments of Martin's Axiom as e.g. in [22, §7] (see also [12,13,23]). …”

“…We notice that if K 0 ∪ K 1 has the rectangle refining property, then so does P, and a suitable refinement is just a ∆-system refinement. For other examples, see [23]. Let K 2 (rec) be the statement that for every two-colored partition on [ω 1 ] 2 with the rectangle refining property, there exists an uncountable K 0 -homogeneous set (defined in [14]), and let MA ℵ 1 (rec) be Martin's Axiom for ℵ 1 -many dense sets of forcing notions with the rectangle refining property.…”

Uniformizing ladder system colorings and the rectangle refining property

“…It is easy to see that a specializing Aronszajn tree by finite approximations has the rectangle refining property in the sense of [14]. The definition in [14] includes that of the present paper (for other examples, see [12,13,15]). …”

“…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.…”