The Cichon diagram

We conclude the discussion of additivity, Baire number, uniformity and covering for measure and category by constructing the remaining 5 models. Thus we complete the analysis of Cichon's diagram

The higher Cichon diagram in the degenerate case

For a regular uncountable cardinal kappa, we discuss the order relationship between the unbounding and dominating numbers on kappa and cardinal invariants of the higher meager ideal M_kappa. In particular, we obtain a complete characterization of add(M_kappa) and cof(M_kappa) in terms of cov(M_kappa) and non(M_kappa) and unbounding and dominating numbers, and we provide models showing that there are no restrictions on the value of non(M_kappa) in the degenerate case 2^{ kappa except 2^{<kappa} leq non(M_kappa) leq 2^kappa. The corresponding question for cof(M_kappa) remains open. Our results answer questions of joint work of the author with Brooke-Taylor, Friedman, and Montoya.Comment: 12 page

The Cicho\'n Diagram for Degrees of Relative Constructibility

Following a line of research initiated in \cite{BBNN}, I describe a general framework for turning reduction concepts of relative computability into diagrams forming an analogy with the Cicho\'n diagram for cardinal characteristics of the continuum. I show that working from relatively modest assumptions about a notion of reduction, one can construct a robust version of such a diagram. As an application, I define and investigate the Cicho\'n Diagram for degrees of constructibility relative to a fixed inner model $W$. Many analogies hold with the classical theory as well as some surprising differences. Along the way I introduce a new axiom stating, roughly, that the constructibility diagram is as complex as possible.Comment: 25 pages, 14 figures. Fourth version adds some citations, cutting down on standard proofs and includes some additional observations based on very helpful comments from an anonymous referee. Now accepted at Mathematical Logic Quarterl

Controlling classical cardinal characteristics while collapsing cardinals

Given a forcing notion $P$ that forces certain values to several classical cardinal characteristics of the reals, we show how we can compose $P$ with a collapse (of a cardinal $\lambda>\kappa$ to $\kappa$) such that the composition still forces the previous values to these characteristics. We also show how to force distinct values to $\mathfrak m$, $\mathfrak p$ and $\mathfrak h$ and also keeping all the values in Cicho\'n's diagram distint, using the Boolean Ultrapower method of arXiv:1708.03691 . (In arXiv:2006.09826 , the same was done for the newer Cicho\'n's Maximum construction, which avoids large cardinals.)Comment: Compared to the previous version arXiv:1904.02617v1 , parts have been removed that are included in arXiv:2006.09826 (a construction for 13 characteristics based on the Cichon's maximum construction without large cardinals, arXiv:1906.06608

Computable analogs of cardinal characteristics: Prediction and Rearrangement

There has recently been work by multiple groups in extracting the properties associated with cardinal invariants of the continuum and translating these properties into similar analogous combinatorial properties of computational oracles. Each property yields a highness notion in the Turing degrees. In this paper we study the highness notions that result from the translation of the evasion number and its dual, the prediction number, as well as two versions of the rearrangement number. When translated appropriately, these yield four new highness notions. We will define these new notions, show some of their basic properties and place them in the computability-theoretic version of Cicho\'{n}'s diagram.Comment: 33 pages, 6 figures, thesis chapte

The left side of Cicho\'n's diagram

Using a finite support iteration of ccc forcings, we construct a model of $\aleph_1<\mathrm{add}(\mathcal{N})<\mathrm{cov}(\mathcal{N})<\mathfrak{b}<\mathrm{non}(\mathcal{M})<\mathrm{cov}(\mathcal{M})=\mathfrak{c}$.Comment: Publication 1066 on Shelah's list, 14 page

Combinatorial properties of classical forcing notions

We discuss the effect of adding a single real (for various forcing notions adding reals) on cardinal invariants associated with the continuum (like the unbounding or the dominating number or the cardinals related to measure and category on the real line). For random and Cohen forcing, this question was investigated by Cicho'n and Pawlikowski; for Hechler forcing, by Judah, Shelah and myself. We show here: (1) adding an eventually different or a localization real adjoins a Luzin set of size continuum and a mad family of size omega_1; (2) Laver and Mathias forcing collapse the dominating number to omega_1 --- consequences: (A) CON(d=omega_1 + unif(L) = unif (M) = kappa = 2^omega) for any regular uncountable kappa; (B) Two Laver or Mathias reals added iteratively always force CH (even diamond); (C) Sigma^1_4-Mathias-absoluteness implies the Sigma^1_3- Ramsey property; (3) Miller's rational perfect set forcing preserves the axiom MA(sigma-centered)

Filter-linkedness and its effect on preservation of cardinal characteristics

We introduce the property ``$F$-linked'' of subsets of posets for a given free filter $F$ on the natural numbers, and define the properties ``$\mu$-$F$-linked'' and ``$\theta$-$F$-Knaster'' for posets in a natural way. We show that $\theta$-$F$-Knaster posets preserve strong types of unbounded families and of maximal almost disjoint families. Concerning iterations of such posets, we develop a general technique to construct $\theta$-$\mathrm{Fr}$-Knaster posets (where $\mathrm{Fr}$ is the Frechet ideal) via matrix iterations of ${<}\theta$-ultrafilter-linked posets (restricted to some level of the matrix). This is applied to prove consistency results about Cicho\'n's diagram (without using large cardinals) and to prove the consistency of the fact that, for each Yorioka ideal, the four cardinal invariants associated with it are pairwise different. At the end, we show that three strongly compact cardinals are enough to force that Cicho\'n's diagram can be separated into $10$ different values.Comment: 30 pages, 7 figure

A note on "Another ordering of the ten cardinal characteristics in Cicho\'n's Diagram" and further remarks

In this note, we relax the hypothesis of the main results in Kellner-Shelah-T\v{a}nasie's "Another ordering of the ten cardinal characteristics in Cicho\'n's diagram".Comment: 15 pages, 1 figure. To appear in Kyoto Daigaku Surikaiseki Kenkyujo Kokyurok

Some models produced by 3D iterations

We construct models, by three-dimensional arrays of ccc posets, where many classical cardinal characteristics of the continuum are pairwise different.Comment: 13 pages, RIMS Set Theory Workshop: Infinite Combinatorics and Forcing Theory. November 28th - December 1st, 2016, Kyoto University, Japan. To appear in Kyoto Daigaku Suurikaiseki Kenkyuusho Koukyuurok