# Infinite Combinatorics and Forcing Theory

## at the Research Institute for Mathematical Sciences, Kyoto University

Tutorial Talks
• Natasha Dobrinen (University of Denver)
Ramsey theory in forcing

This tutorial focuses on certain developments and applications of Ramsey theory to problems in forcing, and conversely, development of forcing to solve problems in Ramsey theory. We will introduce the fundamental ideas in topological Ramsey space theory as well as some essential examples. Then the tutorial will focus on internal topological Ramsey spaces; that is, topological Ramsey spaces which are dense inside given forcings. The motivation for those constructions of such dense subsets is several-fold including the development of canonical equivalence relations on barriers and applying them to obtain exact Tukey and Rudin-Keisler structures. We shall also show how forcing can motivate the development of new Ramsey theorems, as has been the case for proving that certain creature forcings have dense subsets which are topological Ramsey spaces. Finally, we will finish by highlighting some uses of forcing to prove Ramsey theorems as well as survey some recent results and open problems on the interface of Ramsey theory and forcing.
slides;Day1, slides;Day2, slides;Day3

• Yinhe Peng (University of Toronto)
Negative partition relations on $\omega_1$

The history of negative partition relations back to Sierpiński. He constructed a partition of $[ \omega_1 ]^2$ into 2 colors such that every uncountable square contains all 2 colors. It was later improved to 3 colors by Laver, to 4 colors by Galvin and Shelah and then to $\omega_1$ colors by Todorcevic. This is not the end of the story. People start to look at stronger statement -- strong negative partition relations.
Definition. $Pr_0(\kappa, \theta, \sigma)$ asserts that there is a function $c:[\kappa]^2\rightarrow \theta$ such that whenever we are given $\gamma<\sigma$, a collection $\langle a_\alpha: \alpha< \kappa\rangle$ of pairwise disjoint elements of $[\kappa]^{\gamma}$ and a function $h: \gamma\times \gamma\rightarrow \theta$, then there are $\alpha< \beta$ such that $c( a_\alpha(i), a_\beta(j))=h(i,j)$ for any $i,j <\gamma$.
Shelah then proved that $Pr_0(\kappa^+,\kappa^+,\omega)$ holds for any uncountable regular $\kappa$. However, the situation for $Pr_0(\omega_1,\theta,\sigma)$ is different. $Pr_0(\omega_1,2,\omega)$ is not a consequence of ZFC since it implies the non-productivity of ccc forcings. We will see situations on $\omega_1$ and applications. We will also see related problems and further researches.
Chalk talk

Talks
• Jörg Brendle (Kobe University)
Maximal trees

We consider trees (ordered downward) in ${\cal P} (\omega)$ and in ${\cal P} (\omega) / {\mathrm{fin}}$ which are maximal in the sense that they have no proper end-extensions. Monk observed that there are such maximal trees in ${\cal P} (\omega)$ of size $\omega$ and ${\mathfrak c}$, and in ${\cal P}(\omega) / {\mathrm{fin}}$, of size ${\mathfrak c}$, and asked whether there can consistently be maximal trees of other sizes. This was answered by Campero, Cancino, Hrŭsák, and Miranda who showed that one of the parametrized diamond principles implies the existence of a maximal tree of height and width $\omega_1$ in ${\cal P} (\omega) / {\mathrm{fin}}$ and of a tree of height $\omega$ and width $\omega_1$ which is maximal as a subtree of both ${\cal P} (\omega)$ and ${\cal P} (\omega) / {\mathrm{fin}}$. If we define the tree number ${\mathfrak{tr}}$ as the least size of a maximal tree in ${\cal P} (\omega) / {\mathrm{fin}}$, this implies the consistency of $\omega_1 = {\mathfrak{tr}} < {\mathfrak c}$ while the consistency of $\omega_1 < {\mathfrak{tr}} = {\mathfrak c}$ is easy to see.
Answering questions of Campero, Cancino, Hrŭsák, and Miranda, we show that $\omega_1 < {\mathfrak{tr}} < {\mathfrak c}$ is consistent as well, by devising a forcing adding a maximal tree in ${\cal P} (\omega) / {\mathrm{fin}}$ of height and width a prescribed regular uncountable cardinal $\kappa$ over a model of large continuum. By simultaneously adding maximal trees of various sizes we can also obtain large tree spectrum. Moreover, by modifying the construction, we can obtain consistently trees of width $\kappa$ and height $\omega$ which are maximal in both ${\cal P} (\omega) / {\mathrm{fin}}$ and ${\cal P} (\omega)$. Finally, we prove in ZFC that there are no maximal trees in ${\cal P} (\omega) / {\mathrm{fin}}$ of countable width. In my talk, I will present an overview and sketch some of the proofs.
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• Fabiana Castiblanco (Universität Münster)
Preserving sharps by tree forcings

The Levy-Solovay theorem states that if $\mathbb{P}$ is a forcing notion of size $<\kappa$ and $\kappa$ is a measurable cardinal in $V$ then $\kappa$ remains to be measurable in $V^\mathbb{P}$. There are variants to this theorem stating that compact, supercompact, strong and Woodin cardinals are preserved after doing "small" forcing. Therefore, it is natural to ask whether small forcing notions preserves consequences of large cardinals, for instance the existence of sharps.
In this talk, we will show that various tree forcing notions such as Sacks, Silver, Mathias, Miller and Laver preserve sharps for reals. Furthermore, all these forcing notions preserve the existence of $M_n^\#(x)$ for every $n<\omega$, $x\in\mathbb{R}$ which is equivalent to say that all these forcing notions preserve $\mathbf{\Pi^1_n}$-Determinacy according with a result of Martin-Neeman-Steel-Woodin. This is a joint work with Philipp Schlicht.
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• David Chodounsky (Institute of Mathematics of the Czech Academy of Science)
Mathias-Prikry forcing and generic ultrafilters

I will analyze the Mathias--Prikry forcing $M(\mathcal F)$ where $\mathcal F$ is a filter on $\omega$ and derive a condition for reals which is sufficient for genericity in this poset. As an application, I will present a characterization of certain ultrafilters on $\omega$ generic over $L(\mathbb R)$. Joint work with J. Zapletal.
Theorem. Suppose there are large cardinals. Let $\mathcal I$ be an F$_\sigma$ ideal on $\omega$ and $\mathcal U$ be an ultrafilter on $\omega$. The following are equivalent:
1. $\mathcal U$ is $Q$-generic over $L(\mathbb R)$ where $Q$ is the poset of $\mathcal I$-positive sets ordered by modulo finite inclusion,
2. $\mathcal U$ is a P-point, $\mathcal U \cap \mathcal I = \emptyset$, and whenever $E \subset \mathcal P(\omega)$ is a closed set disjoint from $\mathcal U$, then there is a set $e \in \mathcal P(\omega) \setminus \mathcal U$ such that $E \subset \langle \mathcal I , \{\, e \,\} \rangle$.

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• Sakaé Fuchino (Kobe University)
Reflection theorems on non-existence of orthonormal bases of pre-Hilbert spaces

Let $K=\mathbb{R}$ or $\mathbb{C}$. An inner-product space over $K$ (with the inner product $(\mathbb{x},\mathbb{y})\in K$ for $\mathbb{x}$, $\mathbb{y}\in X$) is also called a pre-Hilbert space (over $K$). An orthonormal system $B\subseteq X$ is called an orthonormal basis if $B$ spans a dense subspace of $X$. Under the lack of completeness and separability of the space, a maximal orthonormal system $B$ of $X$ need not to be an orthonormal basis. Halmos proved in 1970s that there are even pre-Hilbert spaces without any orthonormal bases (see [3]).

For an infinite set $S$, let

$\ell_2(S)=\{\mathbb{u} \in{}^{S}{K}:\sum_{x\in S}(\mathbb{u} (x))^2<\infty\}$,

where $\sum_{x\in S}(\mathbb{u} (x))^2$ is defined as $\sup\{\sum_{x\in A}(\mathbb{u} (x))^2:A\in[S]^{<\aleph_0}\}$. $\ell_2(S)$ is endowed with a natural structure of inner product space with coordinatewise addition and scalar multiplication, as well as the inner product defined by

$(\mathbb{u},\mathbb{v})=\sum_{x\in S}\mathbb{u}(x)\overline{\mathbb{v}(x)}$ for $\mathbb{u}$, $\mathbb{v}\in\ell_2(S)$.

It is easy to see that $\ell_2(S)$ is a/the Hilbert space of density $|S|$. For $\mathbb{x}\in\ell_2(S)$ for some $S$, the support of $\mathbb{x}$ --- notation: ${\rm supp}(\mathbb{x})$ --- is the set $\{s\in S:\mathbb{x}(s)\not=0\}$. For $S'\subseteq S$, and $X\subseteq\ell_2(S)$ we denote $X\downarrow S'=\{\mathbb{x}\in X:{\rm supp}(\mathbb{x})\subseteq S'\}$.

Proposition 1. Suppose that $\kappa$ is a supercompact cardinal. Then for any pre-Hilbert space $X$ without any orthonormal bases, there are stationarily many sub-inner-product-spaces $Y$ of $X$ of size $<\kappa$ without any orthonormal bases.

Theorem 2. ([1]) Suppose that $\lambda$ is a singular cardinal and $X$ is a pre-Hilbert space which is a dense sub-inner-product-space of $\ell_2(\lambda)$. If $X$ does not have any orthonormal bases then there is a cardinal $\lambda'<\lambda$ such that

(1) $\{u\in[\lambda]^{\kappa^+}:X\downarrow u\mbox{ is a pre-Hilbert space without any orthonormal bases}\}$

is stationary in $[\lambda]^{\kappa^+}$ for all $\lambda'\leq\kappa<\lambda$.

Proof. A straight forward modification of the proof of the main theorem of [4] will do. Eklof [0] proves a similar singular compactness theorem. $\square$

Theorem 3. ([1]) The following are equivalent over ZFC:

 (a) the Fodor-type Reflection Principle; (b)For any regular $\kappa>\omega_1$ and any linear subspace $X$ of $\ell_2(\kappa)$ dense in $\ell_2(\kappa)$, if $X$ does not have any orthonormal bases then        $S_X=\{\alpha<\kappa:X\downarrow \alpha\mbox{ does not have any orthonormal bases}\}$ is stationary in $\kappa$; (c) regular $\kappa>\omega_1$ and any dense sub-inner-product-space $X$ of $\ell_2(\kappa)$, if $X$ does not have any orthonormal bases then        $S^{\aleph_1}_X=\{U\in[\kappa]^{\aleph_1}: X\downarrow U\mbox{ does not have any orthonormal bases}\}$ is stationary in $[\kappa]^{\aleph_1}$.

Proof. (a) $\Rightarrow$ (b) is done by induction on $d(X)$. The singular cardinal step of the induction follows from Theorem 2. The other direction uses the characterization of FRP in term of the total failure of the principle ${\rm ADS}^-$. We shall discuss more about this in the talk. $\square$

The Fodor-type Reflection Principle (FRP) mentioned above is the reflection principle introduced in [2]. One of the equivalent formulations of the principle is the following:

$\rm (FRP)$ $\$ For any regular $\kappa>\omega_1$, any stationary $S\subseteq E^\omega_\kappa$ and any mapping $g:S\rightarrow[\kappa]^{\aleph_0}$, there is $\alpha^*\in E^{\omega_1}_\kappa$ such that

 $\rm (*)$ $\$ $\alpha^*$ is closed with respect to $g$ (that is, $g(\alpha)\subseteq\alpha^*$ for all $\alpha\in S\cap\alpha^*$) and, for any $I\in[\alpha^*]^{\aleph_1}$ closed with respect to $g$, closed in $\alpha^*$ with respect to the order topology and with $\sup(I)=\alpha^*$, if $\langle I_\alpha:\alpha\lt\omega_1\rangle$ is a filtration of $I$ then $\sup(I_\alpha)\in S$ and $g(\sup(I_\alpha))\cap\sup(I_\alpha)\subseteq I_\alpha$ hold for stationarily many $\alpha\lt\omega_1$.

In the talk I shall also discuss the relationship of the FRP (= the reflection theorem about the non existence of orthonormal bases) with other known reflection assertions.

[0] Paul C. Eklof, Shelah's singular compactness theorem, Publicacions Matemàtiques, Vol.52, No.1, (2008), 3--18.
[1] Sakaé Fuchino, Pre-Hilbert spaces without orthonormal bases, submitted. (arXiv:1606.03869v2)
[2] Sakaé Fuchino, István Juhász, Lajos Soukup, Zoltan Szentmiklóssy and Toshimichi Usuba, Fodor-type Reflection Principle and reflection of metrizability and meta-Lindelöfness, Topology and its Applications Vol.157, 8 (2010), 1415--1429.
[3] Stanley Gudder, Inner Product Spaces, The American Mathematical Monthly, Vol.81, No.1 (1974), 29--36.
[4] Wilfrid Hodges, In singular cardinality, locally free algebras are free, Algebra Universalis, 12, (1981), 205--220.
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• Daisuke Ikegami (Tokyo Denki University)
Boolean valued second order logic

In second order logic, there are two major semantics; one is full semantics (Tarski semantics), and the other is Henkin semantics. While full semantics can express many properties such as well-foundedness of a partial order, it does not satify good properties of logics such as comppleteness theorem. On the other hand, Henkin semantics enjoys many good properties of logics, but it is essentially the same as the standard semantics for first order logic.
In this talk, we introduce Boolean valued semantics for second order logic. Then we investigate several properties of this semantics such as the compactness number, the complexity of validity notion, and the inner model obtained by Gödel's construction of L, and compare them with those of full semantics. This is joint work with Jouko Väänänen.
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• Diego A. Mejía (Shizuoka University)
The left side of Cichoń's diagram

It is well known that, with finite support iterations of ccc posets, we can obtain models where 3 or more cardinals of Cichoń's diagram can be separated. For example, concerning the left side of Cichoń's diagram, it is consistent that $\mathrm{add}(\mathcal{N}) < \mathrm{cov}(\mathcal{N}) < \mathfrak{b} < \mathrm{non}(\mathcal{M}) = \mathrm{cov}(\mathcal{M}) = \mathfrak{c}$. Nevertheless, getting the additional strict inequality $\mathrm{non}(\mathcal{M}) < \mathrm{cov}(\mathcal{M})$ is a challenge because subposets of $\mathbb{E}$, the standard ccc poset that adds an eventually different real, may add dominating reals (Pawlikowski 1992).
We construct a model of $\mathrm{add}(\mathcal{N}) < \mathrm{cov}(\mathcal{N}) < \mathfrak{b} < \mathrm{non}(\mathcal{M}) < \mathrm{cov}(\mathcal{M}) = \mathfrak{c}$ with the help of chains of ultrafilters that allow to preserve certain unbounded families.
This is a joint work with Martin Goldstern and Saharon Shelah.
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• Tadatoshi Miyamoto (Nanzan University)
Forcing a Morass by Finite Side Conditions
Chalk talk
• Diana Carolina Montoya (Kurt Gödel Research Center)
Coherent systems of finite support iterations

In joint work with Vera Fischer, Sy-David Friedman and Diego Mejía we present a generalization of the method of matrix iterations introduced by Blass and Shelah in [Blass and Shelah, Ultrafilters with small generating sets, Israel J. Math. 65.3 (1989), pp. 259–271] which provides generic extensions where many constellations in Cichoń’s diagram can be decided. In particular, we present a model where 7 cardinals can be separated and additionally we present a preservation result that allows us to decide the cardinal invariant a in such models. Specifically we show how to preserve canonical Hechler-style maximal almost disjoint families added at some stage of a particular ccc finite support iteration.
I will give a review of the abstract notion of coherent systems and show the results on the diagram as applications of this general framework.
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• André Ottenbreit Maschio Rodrigues (Kobe University)
On some downwards transfer properties on Foreman-Laver model

This is a joint work with Saka\'e Fuchino and Hiroshi Sakai.
Foreman and Laver proved in [Foreman and Laver. Some downwards transfer properties for $\aleph_2$. Adv. in Math., 67(2):230–238, 1988] the following transfer property from $\aleph_2$ to $\aleph_1$:
Theorem (Laver and Foreman, 1988). Assuming the consistency of ZFC together with the existence of a huge cardinal, there is a model of ZFC+GCH in which the following holds: "For any graph of size and chromatic number $\aleph_2$ there is an induced subgraph of size and chromatic number $\aleph_1$."
This construction is done using a technique by Kunen [Kunen. Saturated ideals. J. Symbolic Logic, 43(1):65–76, 1978], originally used to prove the consistency of an $\aleph_2$-saturated ideal on $\aleph_1$.
In 1994, Tall [Tall. Topological applications of generic huge embeddings. Trans. Amer. Math. Soc., 341(1):45–68, 1994] used the same model by Foreman and Laver to prove the consistency of some downwards transfer properties of $\aleph_2$ related to some topological properties, such as collectionwise Hausdorfness and normality.
We give a survey on the known results on this subject and then present some new developments.
slides (update, April 10, 2017)

• Hiroshi Sakai (Kobe University)
On strong negation of FRP

The Fodor-type Refleciton Principle (FRP) is a set theoretical reflection principle which has many equivalent reflection principles in terms of general topology, infinite graph theory, functional analysis, etc. In this talk we introduce "strong negation of FRP" and discuss its consistency and relationships with other combinatorial principles.
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• David Schrittesser (University of Copenhagen)
Infinite combinatorics, definability and forcing

One of the first studies of definable maximal discrete sets was Arnold W. Miller famous paper Infinite combinatorics and definability''. Several results obtained there can be seen as showing that for various well-known relations $R$ on Baire sace, there exist co-analytic maximal $R$-discrete sets (in an appropriate sense) in the constructible universe $\mathbf{L}$.
Recently, new developments have occurred in this area: On one hand, Shelah and Horowitz have shown that contrary perhaps to expectations, there is a Borel maximally eventually different family---which is unusual as for many relations, a maximal $R$-discrete set cannot be analytic. They have also shown that the non-existence of a reasonably definable $R$-maximal discrete set has large cardinal strength, for some relation $R$, but not when $\neg R$ is almost disjointness''.
Another question is whether discrete sets of low definitional complexity can exist in forcing extensions of $\mathbf{L}$: In joint work with Fischer and Törnquist, we have found a co-analytic maximal cofinitary group in $\mathbf{L}$ which remains maximal after adding any number of Cohen reals; in joint work with Törnquist, we have shown there is a $\Delta^1_2$ maximal $R$-discrete set in the Sacks extension of $\mathbf{L}$ for any analytic binary relation R, and for some well-known relations there is a co-analytic one; later I showed this holds also after adding $\omega_2$ Sacks reals to $\mathbf{L}$. The proofs use Ramsey theory and connect it with forcing in new ways. In this talk, I shall sketch these and some related results.
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• Toshimichi Usuba (Waseda University)
Definability of grounds in ZF

Laver and Woodin independently showed that the ground model is definable in its forcing extensions. Their proof relies on the axiom of choice, so it is natural to ask whether, without the axiom of choice, the ground model is definable in its forcing extension. For this problem, we give a following partial answer: In ZF, suppose there are proper class many large cardinals, then the ground model is definable in its forcing extensions.
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• Jonathan Verner (Charles University)
Ultrafilters and Towers
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• Liuzhen Wu (Chinese Academy of Science)
Hamel basis without well-ordering on $\mathbb R$
slides, coming soon
• Jiachen Yuan (Chinese Academy of Science)
A Geometric condition of extender-based Prikry forcing

Gitik introduced the short extender based forcing whose generic model satisfies the negation of SCH and preserves cofinality of the ground model. The forcing achieves this goal by adding a scale to a countable limit of large cardinals. It is well known that several Prikry type forcings satisfy nice geometric condition ( characterization of generic). In joint work with Liuzhen Wu, we give a characterization of the generic of the gap two short extender forcing. It turns out that the generic can be characterized by a much simpler partial order and the extenders in the ground model. One goal of this project is to yield consistent results of negation of SCH and combinatoric properties with weaker large cardinal assumptions.
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