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 $\mathfrak{r}_{pair}>\mathfrak{b}$.
• Comming soon! soon? 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.

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