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In this paper, we investigate two chain conditions of forcing notions, called the rectangle refining property and property ${\mathsf{R}}_{1,{\mathrm{\aleph }}_1}$. 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.
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 ${\mathsf{R}}_{1,{\mathrm{\aleph }}_1}$ 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.

In this paper, the assertion (c) on a family of club subsets of ${\omega }_1$ will be explained. This is introduced by J. T. Moore, and implies $2^{{\mathrm{\aleph }}_0}\ge {\mathrm{\aleph }}_2$. A family of club subsets of ${\omega }_1$ cannot be be diagonalized in any outer model with the same ${\mathrm{\aleph }}_2$.
We introduce an Asperó-Mota iteration which forces the assertion on (c). By thinking the analogous assertion of (c) on stationary subsets of ${\omega }_1$, the property of the iteration disclose
a technical limitation of some types of Asperó-Mota iterations.

It is proved that YPFA implies MRP.

We introduce a fragment ${\mathsf{P}\mathsf{F}\mathsf{A}}^{\text{s-fin}}({\omega }_1)$ of Asperó-Mota’s finitely proper forcing axiom ${\mathsf{P}\mathsf{F}\mathsf{A}}^{\mathrm{f}\mathrm{i}\mathrm{n}}({\omega }_1)$. ${\mathsf{P}\mathsf{F}\mathsf{A}}^{\text{s-fin}}({\omega }_1)$ implies some consequences of ${\mathsf{P}\mathsf{F}\mathsf{A}}^{\mathrm{f}\mathrm{i}\mathrm{n}}({\omega }_1)$, for example, ${\mathsf{M}\mathsf{A}}_{{\mathrm{\aleph }}_1}$, the failure of $\mho$, no weak club guessing ladder systems, and the assertion that every two Aronszajn trees are club-isomorphic. For each integer $k\ge 2$, it is consistent that ${\mathsf{P}\mathsf{F}\mathsf{A}}^{\text{s-fin}}({\omega }_1)$ holds, there exists a $k$-entangled set of reals, and $2^{{\mathrm{\aleph }}_0}={\mathrm{\aleph }}_2$. This extends Abraham-Shelah’s theorem that Martin’s Axiom does not imply that every two ${\mathrm{\aleph }}_1$-dense sets of reals are isomorphic.

In this paper it is proved that, when $Q$ is a quiver that admits some closure, for any algebraically closed field $K$ and any finite dimensional $K$-linear representation $\mathcal{X}$ of $Q$, if ${\mathrm{E}\mathrm{x}\mathrm{t}}_{KQ}^1(\mathcal{X},KQ)=0$ then $\mathcal{X}$ is projective.In contrast, we show that if $Q$ is a specific quiver of the type above, then there is an infinitely generated non-projective $KQ$-module $M_{{\omega }_1}$ such that, when $K$ is a countable field, ${\mathsf{M}\mathsf{A}}_{{\mathrm{\aleph }}_1}$ (Martin’s Axiom for ${\mathrm{\aleph }}_1$ many dense sets, which is a combinatorial axiom in set theory) implies that ${\mathrm{E}\mathrm{x}\mathrm{t}}_{KQ}^1(M_{{\omega }_1},KQ)=0$.

David Asperó and Miguel Angel Mota discovered a new method of iterated forcing using models as side conditions [Forcing consequences of PFA together with the continuum large, Trans. Amer. Math. Soc. 367 (2015), no. 9, 6103-6129]. The side condition method with models was introduced by Stevo Todorčević in the 1980s. The Asperó-Mota iteration enables us to force some ${\mathrm{\Pi }}_2$-statements over $H({\mathrm{\aleph }}_2)$ with the continuum greater than ${\mathrm{\aleph }}_2$. In this article, by using the Asperó-Mota iteration, we prove that it is consistent that $\mho$ fails, there are no weak club guessing ladder systems, $\mathfrak{p}=\mathrm{add}(\mathcal{N})=2^{{\mathrm{\aleph }}_0}>{\mathrm{\aleph }}_2$ and ${\mathsf{MA}}_{{\mathrm{\aleph }}_1}$ fails.

We show that under $\mathsf{PFA}(S)$, the coherent Suslin tree forces that every two Aronszajn trees are club-isomorphic.

It is proved that ideal-based forcings with the side condition method of Todorčević add no random reals. By applying Judah-Repický’s preservation theorem, it is consistent with the covering number of the null ideal equal to ${\mathrm{\aleph }}_1$ that there are no $S$-spaces, every poset of uniform density ${\mathrm{\aleph }}_1$ adds ${\mathrm{\aleph }}_1$ Cohen reals, there are only five cofinal types of directed posets of size ${\mathrm{\aleph }}_1$, and so on. This extends the previous work due to Zapletal [Keeping additivity of the null ideal small, Proc. Amer. Math. Soc. 125 (1997), no. 6, 2443-2451].
Chodounský and Zapletal developed this result in [Why Y-c.c, Ann. Pure Appl. Logic 166 (2015), no. 11, 1123–1149].

One of the main problems about $\mathsf{PFA}(S)$ is that whether a coherent Suslin tree forces that there are no S-spaces under $\mathsf{PFA}(S)$. We analyze a forcing notion related to this problem, and show that under $\mathsf{PFA}(S)$, $S$ forces that every topology on ${\omega }_1$ generated by a basis in the ground model is not an S-topology. This supplements the previous work due to Stevo Todorčević.

In [Two examples of Borel partially ordered sets with the countable chain condition, Proc. Amer. Math. Soc. 112 (1991), no. 4, 1125-1128], Todorčević introduced a ccc forcing which is Borel definable in a separable metric space. In [On Todorcevic orderings, Fund. Math., 228 (2015), 173-192], Balcar, Pazák and Thümmel applied it to more general topological spaces and called such forcings Todorčević orderings. There they analyze Todorčević orderings quite deeply. A significant remark is that Thümmel solved the problem of Horn and Tarski by use of Todorčević ordering [The problem of Horn and Tarski, Proc. Amer. Math. Soc. 142 (2014), no. 6, 1997-2000].
This paper supplements the analysis of Todorčević orderings due to Balcar, Pazák and Thümmel. More precisely,
it is proved that Todorčević orderings add no random reals whenever they have the countable chain condition.
Chodounský and Zapletal developed this result in [Why Y-c.c, Ann. Pure Appl. Logic 166 (2015), no. 11, 1123–1149].

My contribution is that Jensen’s diamond implies the existence of a Suslin lower semi-lattice which is a substructure of $⟨\mathcal{P}(\omega ),⊊,\cap ⟩$. All of other results are proved by Dilip.

We prove a combinatorial lemma which enables us to prove that Balogh’s natural Dowker space is not countably paracompact without using elementary submodes.

We investigate forcing notions with the rectangle refining property, which is stronger than the countable chain condition, and fragments of Martin’s Axiom for such forcing notions. We prove that it is consistent that every forcing notion with the rectangle refining property has precaliber ${\mathrm{\aleph }}_1$ but ${\mathsf{MA}}_{{\mathrm{\aleph }}_1}$ for forcing notions with the rectangle refining property fails.

We introduce a property of forcing notions, called the anti-${\mathsf{R}}_{1,{\mathrm{\aleph }}_1}$, which comes from Aronszajn trees. This property canonically defines a new chain condition stronger than the countable chain condition, which is called the property ${\mathsf{R}}_{1,{\mathrm{\aleph }}_1}$.In this paper, we investigate the property ${\mathsf{R}}_{1,{\mathrm{\aleph }}_1}$. For example, we show that a forcing notion with the property ${\mathsf{R}}_{1,{\mathrm{\aleph }}_1}$ does not add random reals. We prove that it is consistent that every forcing notion with the property ${\mathsf{R}}_{1,{\mathrm{\aleph }}_1}$ has precaliber ${\mathrm{\aleph }}_1$ and ${\mathsf{MA}}_{{\mathrm{\aleph }}_1}$ for forcing notions with the property ${\mathsf{R}}_{1,{\mathrm{\aleph }}_1}$ fails. This negatively answers a part of one of classical problems about implications between fragments of ${\mathsf{MA}}_{{\mathrm{\aleph }}_1}$.

We introduce a generalization of a Dowker space constructed from a Suslin tree due to Mary Ellen Rudin, and the rectangle refining property for forcing notions, which is modified the one for partitions due to Paul B. Larson and Stevo Todorčević and is a property stronger than the countable chain condition. It is proved that Martin’s Axiom for forcing notions with the rectangle refining property implies that every generalized Rudin’s space from Aronszajn trees is not Dowker, and a Suslin tree may force that every generalized Rudin’s space with Aronszajn trees is not Dowker. Moreover, we observe generalized Rudin’s spaces with some types of
non-Aronszajn ${\omega }_1$-trees under the Proper Forcing Axiom.

We introduce the anti-rectangle refining property for forcing notions and investigate fragments of Martin’s axiom for ${\mathrm{\aleph }}_1$ dense sets related to the anti-rectangle refining property, which is close to some fragment of Martin’s axiom for ${\mathrm{\aleph }}_1$ dense sets related to the rectangle refining property, and prove that they are really weaker fragments.

We investigate the finite support product of forcing notions related to destructible gaps, and prove the existence of a large set of independent destructible gaps under $\mathrm{♢}$.

We prove the iteration lemmata, which are the key lemmata to show that extensions by ${\mathbb{P}}_{\mathrm{m}\mathrm{a}\mathrm{x}}$ variations satisfy absoluteness for ${\mathrm{\Pi }}_2$-statements in the structure for some set of reals in $L(\mathbb{R})$, for the following statements: (1) The cofinality of the null ideal is ${\mathrm{\aleph }}_1$. (2) There exists a good basis of the strong measure zero ideal.

We show that (1) The principle “Stick” plus the covering number of the meager ideal larger than ${\mathrm{\aleph }}_1$ implies the existence of a destructible gap and (2) $\mathrm{♣}$ plus the cofinality of the meager ideal equal to ${\mathrm{\aleph }}_1$ implies the existence of a destructible gap.

I proved the title which answers a question addressed in the paper of Moore-Hrusak-Dzamonja [Parametrized $\mathrm{♢}$ principles, Trans. Amer. Math. Soc. 356 (2004), no. 6, 2281–2306].

Supplementing the well known results of Kunen we show that Martin’s Axiom is not sufficient to decide the existence of $({\omega }_1,\mathfrak{c})$-gaps when $(\mathfrak{c},\mathfrak{c})$-gaps exist, that is, it is consistent with ZFC that Martin’s Axiom holds and there are $(\mathfrak{c},\mathfrak{c})$-gaps but no $({\omega }_1,\mathfrak{c})$-gaps.

This paper has two main results. One is that in the extension with Cohen, random and Hechler forcing notions, the set of old reals is Marczewski null. The other is that in the extension with a finite support iteration of Hechler forcing, the covering number of the Marczewski ideal is ${\mathrm{\aleph }}_1$.

The main results of this paper is that it is consistent with ZFC that the value of the cofinality of the strong measure zero ideal is less than the continuum. In this paper we study the relationship between it’s cofinality and the dominating number in the set of functions from $\kappa$ into $\kappa$.

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”.

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 ${\omega }_1\cap \mathsf{\text{Lim}}$, $\mathsf{\text{U}}(\mathcal{S})$ is the assertion that, for any ladder system coloring $⟨d_{\alpha }:\alpha \in {\omega }_1\cap \mathsf{\text{Lim}}⟩$, there exists $S\in \mathcal{S}$ such that the restricted coloring $⟨d_{\alpha }:\alpha \in S⟩$ can be uniformized. Shelah’s proof can be separated into the following two theorems: ${\mathsf{\text{MA}}}_{{\mathrm{\aleph }}_1}$ implies $\mathsf{\text{U}}(\{{\omega }_1\cap \mathsf{\text{Lim}}\})$, and $\mathsf{\text{U}}(\{{\omega }_1\cap \mathsf{\text{Lim}}\})$ 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 $\mathsf{\text{U}}(\mathsf{\text{stat}})$ holds, where $\mathsf{\text{stat}}$ stands for the set of stationary subsets of ${\omega }_1\cap \mathsf{\text{Lim}}$.

By use of the idea of Bagaria and Shelah in [On partial orderings having precalibre-${\mathrm{\aleph }}_1$ and fragments of Martin’s axiom, Fund. Math. 232 (2016), no. 2, 181–197], it is proved that it is consistent that $\mathsf{MA}(\mathrm{r}\mathrm{e}\mathrm{c})$ holds, and both ${\mathcal{K}}_3^{\prime }$ and ${\mathsf{MA}}_{{\mathrm{\aleph }}_1}(\sigma \text{-linked})$ fail.

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.

It is proved that it is consistent that every forcing notions with ${\mathsf{R}}_{1,{\mathrm{\aleph }}_1}$ has precaliber ${\mathrm{\aleph }}_1$, every Todorčević ordering for any second countable Hausdorff space also has precaliber ${\mathrm{\aleph }}_1$, and there exists a non-special Aronszajn tree. This slightly extends the previous work due to the author [APAL 2010].

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.

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.

We show that if $\mathsf{ZF}$ is consistent with the Axiom of Determinacy, then it is consistent that $N{S}_{{\omega }_1}$ is saturated and there exists a Suslin tree $S$ such that the forcing to specialize $S$ with finite conditions destroys the presaturation of $N{S}_{{\omega }_1}$. Our proof is a different approach by Velickovic [Forcing axioms and stationary sets, Theorem 4.6].

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 ${\mathrm{\aleph }}_1$.