About my research

Around Parametrized Diamond principles

- Diamond principles in Cichon's diagram.

(Archive for Mathematical Logic, 2005, May, Volume 44, Number 4, 513--526)

Some models satisfying Parametrized $\diamondsuit$-principles and CH are presented

by using $\omega_1$ -stage finite support iteration of c.c.c forcing notions.

- Finite support iteration of c.c.c forcing notions and Parametrized $\diamondsuit$-principles

(RIMS Kokyuroku [Forcing method and Large cardinals] (No 1423) pp69-83, April 2005.

By using $\omega_2$ -stage finite support iteration of Suslin forcing notions some models of Parametrized $\diamondsuit$-principles are constructed. - Suslin forcing and Parametrized $\diamondsuit$-principle

(The Journal of Symbolic Logic, 73, 2008, No 3, 752--764 )

By using $\omega_2$ -stage finite support iteration of Suslin forcing notions some models of Parametrized $\diamondsuit$-principles are constructed.

In this paper some technique are developed on the above paper.

Around partitions of $\omega$

- independence number for partitions of omega.

(RIMS Kokyuroku Vol.1530(20070200) pp. 37-48 )

In this paper we define the independence number for partitions of $\omega$.

By using Cohen forcing we will prove the consistency of the cardinal invariant smaller than the continuum.

In this paper you can find much typo. I will revise and develop this topics ...

- Around splitting and reaping number for partitions of $\omega$

(To appear, Archive for Mathematical Logic,Volume 49, Number 4, 501-518).

We investigate splitting number and reaping number for the structure $(\omega)^\omega$ of infinite partitions of $\omega$. - On pair-splitting and pair-reaping

( RIMS Kokyuroku 1595 pp.20-31)

In this paper we investigate variations of splitting number and reaping number,

pair-splitting number $\mathfrak{s}_{pair}$ and pair-reaping number $\mathfrak{r}_{pair}$.

We prove that it is consistent that $\mathfrak{s}_{pair}<\mathfrak{d}$. We also prove it is consistent that $\mathfrak{r}_{pair}>\mathfrak{b}$. - Independent families and reaping families for partitions of $\omega$.

preprint.

In this paper, we will define a cardinal invariant corresponding to the independence number for partitions of $\omega$.

By using forcing along template we will prove this cardinal invariants is larger than a cardinal invariant

corresponding to the reaping number for partitions of $\omega$.

Around ideals on $\omega$

- Around pair-splitting, pair-reaping and cardinal invariants of $F_{\sigma}$-ideals

(with Michael Hrusak and David Meza-Alcantara)

Journal of Symbolic Logic, Volume 75, Issue 2, 661-677

There is no relation to Black Jack.

We investigate the pair-splitting number $\mathfrak{s}_{pair}$ which is a variation of splitting number,

pair-reaping number $\mathfrak{r}_{pair}$ which is a variation of reaping number and cardinal invariants of ideals on $\omega$.

We also study cardinal invariants of $F_{\sigma}$ ideals and their upper bounds and lower bounds.

As an application, we answer a question of S. Solecki by showing that

the ideal of finitely chromatic graphs is not locally $Kat\check{e}tov$-minimal among ideals not satisfying Fatou's lemma.

- Mathias-Prikry and Laver-Prikry type forcing

(with Michael Hrusak)

We study Mathias forcing and Laver forcing associated with filters on $\omega$.

We give a combinatorial characterization of Martin's number for these forcing notions and

show a general scheme for analyzing preservation properties for these forcing notions.

Notes

- Cardinal invariants of ideals on $\omega$.

In preparation. - Mad family inextensible to $F_\sigma$ ideal

Under CH, Laflamme constructs a mad family which is inextensible to every $F_\sigma$ ideal.

He also pointed out a construction of a model which have a mad family inextensible to every $F_\sigma$ ideal without CH

by using Cohen-indestructible mad family.

We will construct a model which has a mad family inextensible to $F_\sigma$-ideal

by using finite support iteration of Mathias-Prikry type forcings.

Slides

- Forcings with the countable chain condition and Parametrized
$\diamondsuit$ principles

used at 33rd Winter School in Abstract Analysis -Section of Topology, Raspenava, January 2005. - Around splitting and reaping number for partitions of $\omega$

used at Logic Colloquium 2007, Wroclaw, Poland. - Independence number for partitions of $\omega$

used at 1st European Set Theory Meeting Bedlewo, Poland, 2007. - On the uniformity number of $F_{\sigma}$-ideals on $\omega$

used at Set Theory in Kasugai 2008, February 2008.

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