# works

Refereed papers

Unrefereed papers

• Todorčević’s Axiom $\mathcal K_2$ and ladder system colorings (RIMS 2019), RIMS Kokyuroku, No.2164 (2020), 104-107.(Joint work with Justin Tatch Moore)

In this article, it is proved that if every c.c.c. partition $K \subseteq [\omega_1]^2$ has an ncountable homogeneous set, then every ladder system coloring on $\omega_1$ can be $\sigma$-uniformized. This improves a previous result of the second author in “Todorčević’s fragments of Martin’s axiom and variations of uniformizations of ladder system colorings”.

• Todorčević’s fragments of Martin’s axiom and variations of uniformizations of ladder system colorings (RIMS 2018), RIMS Kokyuroku, No. 2141 (2019), 49-55.

Uniformization of ladder system colorings has been introduced by analysis of a proof of the Shelah’s solution of Whitehead problem. Here, for a subset $\mathcal{S}$ of $\omega_1\cap \textsf{Lim}$, $\textsf{U}(\mathcal{S})$ is the assertion that, for any ladder system coloring $\langle d_\alpha : \alpha \in \omega_1 \cap \textsf{Lim} \rangle$, there exists $S \in \mathcal{S}$ such that the restricted coloring $\langle d_\alpha : \alpha \in S \rangle$ can be uniformized. Shelah’s proof can be separated into the following two theorems: $\textsf{MA}_{\aleph_1}$ implies $\textsf{U}(\{\omega_1 \cap \textsf{Lim}\})$, and $\textsf{U}(\{\omega_1 \cap \textsf{Lim} \})$ implies the existence of a non-free Whitehead group. It is proved that the assertion $\mathcal{K}_3$, which is one of Todorcevic’s fragments of Martin’s Axiom, implies that $\textsf{U}(\textsf{stat})$ holds, where $\textsf{stat}$ stands for the set of stationary subsets of $\omega_1 \cap \textsf{Lim}$.

• A comment on Bagaria-Shelah’s fragment of Martin’s axiom, Iterated forcings and cardinal invariants (RIMS 2017), RIMS Kokyuroku, No. 2081 (2018), 62-68.

By use of the idea of Bagaria and Shelah in [On partial orderings having precalibre-$\aleph_1$ and fragments of Martin’s axiom, Fund. Math. 232 (2016), no. 2, 181–197], it is proved that it is consistent that $\mathsf{MA}({\rm rec})$ holds, and both $\mathcal{K}_3’$ and $\mathsf{MA}_{\aleph_1}(\sigma\text{-linked})$ fail.

• No Suslin trees but a non-special Aronszajn tree exists by a side condition method : compact version, Iterated forcings and cardinal invariants (RIMS 2017), RIMS Kokyuroku, No. 2081 (2018), 28-40. (Joint work with Tadatoshi Miyamoto)

Shelah proved that it is consistent that Suslin Hypothesis holds and there exists a non-special Aronszajn tree. We show this by use of a modification of Aspero-Mota iteration.

• The existence of a non special Aronszajn tree and Todorcevic orderings, Infinitary combinatorics in set theory and its applications (RIMS 2014), Sūrikaisekikenkyūsho Kōkyūroku, No 1949 (2015), 89-98.

It is proved that it is consistent that every forcing notions with $\mathsf R_{1,\aleph_1}$ has precaliber $\aleph_1$, every Todorčević ordering for any second countable Hausdorff space also has precaliber $\aleph_1$, and there exists a non-special Aronszajn tree. This slightly extends the previous work due to the author [APAL 2010].

• Some results in the extension with a coherent Suslin tree II, Forcing extensions and large cardinals (RIMS 2012), Sūrikaisekikenkyūsho Kōkyūroku, No 1851 (2013), 49-61. (Joint work with Tadatoshi Miyamoto)

It is proved the following: (1) $\mathsf{PFA}(S)$ implies $\mathsf{MRP}$. (2) For any coherent Suslin tree $S$ and an $S$-name $\dot{C}$ for a ladder system on $\omega_1$, there exists an almost strongly proper forcing notion (which will be defined in this article) which generically adds an $S$-name for a club on $\omega_1$ which violates the statement that $\dot{C}$ is a weak club guessing ladder systems.

• Some results in the extension with a coherent Suslin tree, Aspects of Descriptive Set Theory (RIMS 2011), Sūrikaisekikenkyūsho Kōkyūroku, No 1790 (2012), 72-82. (Joint work with Dilip Raghavan)

We show that under $\mathsf{PFA}(S)$, the coherent Suslin tree $S$ (which is a witness of the axiom $\mathsf{PFA}(S)$) forces that there are no $\omega_2$-Aronszajn trees. We also determine the values of cardinal invariants of the continuum in this extension.

• Another c.c.c. forcing that destroys presaturation, Combinatorial set theory and forcing theory (RIMS2009), Sūrikaisekikenkyūsho Kōkyūroku, No. 1686 (2010), 73-74. (joint work with Paul B. Larson)

We show that if $\mathsf{ZF}$ is consistent with the Axiom of Determinacy, then it is consistent that $NS_{\omega_{1}}$ is saturated and there exists a Suslin tree $S$ such that the forcing to specialize $S$ with finite conditions destroys the presaturation of $NS_{\omega_{1}}$. Our proof is a different approach by Velickovic [Forcing axioms and stationary sets, Theorem 4.6].

• The inequality $\mathfrak{b} > \aleph_1$ can be considered as an analogue of Suslin’s Hypothesis, Axiomatic Set Theory and Set-theoretic Topology (RIMS 2007), Sūrikaisekikenkyūsho Kōkyūroku, No. 1595 (2008), 84-88.

The author introduced a new chain condition, called the anti-rectangle refining property. A typical example of a forcing notion with the anti-rectangle refining property is an Aronszajn tree. So it can be considered a generalization of Suslin’s Hypothesis that every forcing notion with the anti-rectangle refining property has
an uncountable antichain. It is proved that this statement implies that the bounding number is larger than $\aleph_1$.

Submitted papers

• Two chain conditions and their Todorčević’s Fragments of Martin’s Axiom
• Aspero-Mota iteration and the size of the continuum