Set Theory: Reals and Topology
集合論：連続体上の組合せ論と位相空間論
November 16th (Mon.)〜 20th (Fri.), 2020

Last update: November 12th, 2020

RIMS Workshop 2020
Set Theory: Reals and Topology
Program
Time in Japan Standard Time (GMT +9)

Mon.

Opening

Guzmán (Mini course)

ParenteKumar (Mini course)

SchilhanFischer

Nov. 16th

9:00-9:10

9:10-10:10

10:40-11:20

14:00-15:00

15:25-15:55

16:10-16:50

Tue.

Martínez-Ranero

Ishiu

Lambie-Hanson

Kumar (Mini course)

Eskew

Agostini

Nov. 17th

9:00-9:30

9:45-10:15

10:30-11:00

14:00-15:00

15:25-15:55

16:10-16:50

Wed.

Guzmán (Mini course)

Jirattikansakul

Yoshinobu

9:00-10:00

10:30-11:00

11:15-11:35

Nov. 18thThu.
Nov. 19th
Fri.
Nov. 20th

Ramos-García (Invited)

Sakai

Kumar (Mini course)

Cardona

Supina9:00-10:00

10:30-11:10

14:00-15:00

15:25-15:55

16:10-16:50

Guzmán (Mini course)

Sonpanow

FuchinoIllanelaGoldstern (Invited)

9:00-10:00

10:30-11:00

14:00-14:40

15:05-15:35

15:50-16:50

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Tutorials

Osvaldo GuzmánSome Constructions of MAD families
A MAD family A is called completely separable if every set that is not in the ideal
generated by A, contains an element of A. A famous problem of Erdos and Shelah asks ifsuch MAD family can be constructed in ZFC alone. In his 935 paper, Shelah developed
a novel method to construct MAD families with special properties. In that paper, heconstructed a completely MAD family under diﬀerent assumptions (providing an ”almost”
ZFC result). This method has been simplified and expanded by Mildenberger, Raghavan,and Steprans. In this tutorial, we will explain this method to construct special MADfamilies and some of its applications.

Ashutosh Kumar
Forcing with sigma ideals and Lebesgue outer measure
We will discuss some applications of forcing and generic ultrapowers to Ramsey-type
problems about Lebesgue outer measure.

2Invited talks

Martin Goldstern
Many cardinals below the continuum
It is well known that “the continuum can be anything it ought to be”, and in particular,that the interval from ℵ1 to c = 2ℵ0 can contain arbitrarily many cardinals. It is also an
old result that there can be many cardinals in this interval which all have particularly
simple definitions.
I my talk I will sketch the construction of a ZFC model where many of the well known
cardinal characteristics of the continuum (in particular, the cardinals from Cichoń’s diagram, and some others such as p and s) can be “as distinct as they ought to be”.
The construction uses a finite support iteration of small forcing notions. The mostrecent element of the construction (which is responsible for the value of the splitting
number s) exploits the homogeneity of the forcings we use.
This is joint work with Jakob Kellner, Diego Mejı́a and Saharon Shelah.

1Ulises Ariet Ramos-Garcı́aOCA in the class of hereditarily Lindelöf submetrizable spaces
The purpose of this talk is to analyze Todorcevic’s Open Coloring Axiom (OCA) inthe class of hereditary Lindelöf submetrizable spaces. This class naturally extends to theclass of all separable metrizable spaces where OCA is typically defined. Among otherresults, we show that the definable version for OCA (OCA*) holds for the class of allhereditary Lindelöf strong Choquet submetrizable spaces. In particular, OCA* holds
for the Sorgenfrey line and for the space (ω ω , τGH ) where τGH is the Gandy-Harrington
topology on ω ω .
This is joint work with José Antonio Corona-Garcı́a.

3Contributed talks

Claudio Agostini
A characterization of finite Ramsey monoids
Joint work with Eugenio Colla.
Carlson’s theorem on variable words and Gowers’ F INK theorem are generalizationsof Hindman’s theorem that involve a monoid action on a semigroup. In short, they statethat for any finite coloring of a semigroup there is an infinite monochromatic “span”.
They diﬀer in the choice of the monoid.
Recently, Solecki isolated from these two theorems the notion of Ramsey monoid, providing a common generalization of them. Then he proved that an entire class of finite
monoids is Ramsey.
In this talk, I will present some of the result from a joint work with Eugenio Colla,
where we prove a generalization of Solecki’s theorem, enlarging the class of monoids thatcan be proved to be Ramsey and reaching a simple algebraic characterization of finite
Ramsey monoids.

Miguel A. Cardona
A new result from the creature world
In this talk we use a countable support construction of creature posets to force that there
are continuum many pairwise diﬀerent cardinal invariants where continuum many are ofeach of the following types: covering numbers associated with Yorioka ideals; uniformity
numbers associated with a Yorioka ideals; localization cardinals; and anti-localization
cardinals.
This is a joint work with Diego Mejı́a and Lukas D. Klausner.

2Monroe Eskew
Weak square from weak presaturation
Can we have both a saturated ideal and the tree property on ℵ2 ? Towards the negative
direction, we show that for a regular cardinal κ, if 2<κ ≤ κ+ and there is a weaklypresaturated ideal on κ+ concentrating on cofinality κ, then □∗κ holds. This partiallyanswers a question of Foreman and Magidor about the approachability ideal on ℵ2 . Asurprising corollary is that if there is a weakly presaturated ideal J on ℵ2 such thatP (ℵ2 )/J is a proper forcing, then CH holds. This is joint work with Sean Cox.

Vera Fischer
The spectrum of independence
In this talk, we will show that the set of cardinalities of maximal independent families,
referred to as the spectrum of independence, can be quite arbitrary. In particular, we willshow that every infinite subset of {ℵn }1≥n∈ω can be realised (in a generic extension) as
the spectrum of independence. This is a joint work with S. Shelah.

Sakae Fuchino
Rado’s Conjecture and Hamburger’s Hypothesis
Hamburger’s Hypothesis HH(< 2ℵ0 ) with the reflection point “< 2ℵ0 ” is known to beindependent over a combination of other strong reflection principles where the combination
includes Rado’s Conjecture RC(≤ 2ℵ0 ) with the reflection point “≤ 2ℵ0 ” as well as thePκ (λ) version of the stationarity reflection with the reflection point < 2ℵ0 which implies
that the continuum is fairly large. (see [1]).
In this talk, we show that we can refine the method in [1] to show that HH(< 2ℵ0 ) is
independent over other reflection principles including RC(< 2ℵ0 ).
We also show that Rado’s Conjecture RC (with the reflection point ≤ ℵ1 ) does not
imply Hamburger’s Hypothesis HH (with the reflection point ≤ ℵ1 ) whose consistency isa persistently open problem.
[1] Sakaé Fuchino, André Ottenbreit Maschio Rodrigues and Hiroshi Sakai, Strong downward Löwenheim-Skolem theorems for stationary logics, III — mixed support iteration,
to appear in the Proceedings of the Asian Logic Conference 2019.
Extended version of the paper:https://fuchino.ddo.jp/papers/SDLS-III-xx.pdf

3Martina Iannella
The complexity of convex bi-embeddability between countable linear orders
and some applications
Consider the set LO of countable linear orders and the following relation of “convex
embeddability among them:
L ⊴LO M iﬀ L is isomorphic to a convex set in M.
One easily gets that ⊴LO is an analytic quasi-order on the Polish space LO. We show
that, in contrast to the usual embeddability between linear orders, the relation ⊴LO iscombinatorially complicated: it is not a well quasi-order (wqo). Indeed, it has both innite
descending chains and antichains of size continuum. Denote by ▷◁LO the equivalence
relation on LO induced by ⊴LO .
Theorem. (i) The isomorphism relation ∼=LO between linear orders is Borel reducible
to ▷◁LO . In particular, ▷◁LO is a proper analytic equivalence relation.
(ii) ▷◁LO is not complete for analytic equivalence relations.
Moreover, we have evidence that ▷◁LO is not much more complicated than ∼=LO . Indeed,
we found a partition of LO in four subsets such that the restriction of ▷◁LO on each ofthem is Borel reducible to ∼=LO .
As an application, after reviewing a recent result of Kulikov on the classification of wild
proper arcs/knots up to equivalence [1], we consider a binary relation called “component
which is crucial to that proof. Since such relation turns out to be a quasi-order, we consider
the problem of classifying (wild) arcs up to the associated equivalence relation and weshow that the complexity of such relation is bounded from below by ▷◁LO .
[1] V. Kulikov, A Non-classification Result for Wild Knots, Transactions of theAmerican Mathematical Society, vol. 369 (2017) n. 8, pp. 5829–5853.

Tetsuya Ishiu
There is no Peano Curve for a compact connected nowhere separable linearlyorder topological space
We shall show that if L is a compact connected nowhere separable linearly ordered
topological space, then there is no continuous surjection from L onto the product of two
copies of L. It means that the existence of space-filling curves such as the Peano Curveis specific to the real interval.

4Sittinon Jirattikansakul
Blowing up singular cardinal of uncountable cofinality with collapses
One of the main combinatorial principles in set theory is singular cardinal hypothesis(SCH). It was known by Gitik that to violate SCH, large cardinal assumption is required.
Recently Gitik built up a forcing to violate SCH at any given cofinality, assuming thecardinal is singular in the ground model. In this talk, I will present a similar forcing withcollapses to violate SCH at small cardinals with the same setting.

Chris Lambie-Hanson
Highly connected Ramsey theory on the real numbers
In recent work, Bergfalk, Hrusak, and Shelah introduce some natural nontrivial weakenings of the classical partition relation µ → (µ)2κ that can consistently hold for uncountable
but non-weakly compact cardinals µ. Roughly speaking, these weakenings replace the assertion of the existence of large monochromatic complete subgraphs in the classical partition relation with the assertion of the existence of large highly connected monochromaticsubgraphs. We present some recent work on this topic, including some results about theinfluence of PFA and square principles on these partition relations. Our main result is a
sharp consistency result indicating that a square-bracket version of this partition relation
can consistently hold at the continuum, and that this is in fact equiconsistent with theexistence of a weakly compact cardinal. Time permitting, we will discuss some potential
higher-dimensional generalizations of these results.

Carlos Martı́nez-Ranero
Hereditary interval algebras and cardinal characteristics of the continuum
An interval algebra is a Boolean algebra which is isomorphic to the algebra of finite
unions of half-open intervals, of a linearly ordered set. An interval algebra is hereditary
if every subalgebra is an interval algebra. We answer a question of M. Bekkali and S.
Todorevi, by showing that it is consistent that every σ-centered interval algebra of sizeb is hereditary. We also show that there is, in ZFC, an hereditary interval algebra ofcardinality ℵ1 .

Francesco Parente
Combinatorics of ultrafilters on Boolean algebras
I will present recent joint work with Jörg Brendle, concerning the structure of ultrafilters on Boolean algebras in the framework of Tukey reducibility. The main motivation isto understand and classify (not necessarily generic) ultrafilters on the algebras adjoining
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Cohen and random reals. Furthermore, I will highlight the connection with a cardinalinvariant of Boolean algebras, the ultrafilter number, and prove consistency results concerning its possible values.

Hiroshi Sakai
Higher stationary reflection and cardinal arithmetic
The Weak Reflection Principle, WRP, is known to have many interesting consequences.
Among other things, it implies that 2ω ≤ ω2 and the Singular Cardinal Hypothesis holds.
In this talk, we discuss higher analogs of WRP. We study their consequences on cardinalarithmetic.

Nattapon Sonpanow
Cardinal characteristics associated with families of functions and permutations
A cardinal characteristic of the continuum is a cardinal that lies between ℵ1 and cwhich describes a combinatorial property of the continuum. These cardinals are mostly
defined on families of infinite sets of natural numbers. We study cardinal characteristics
associated with families of functions and permutations on the set of natural numberswith some combinatorial properties. We give relations among these cardinals and otherwell-known ones.

Jonathan Schilhan
Definability of maximal families of reals in forcing extensions
We study the definability of maximal independent sets in analytic hypergraphs. This
includes many of the classical types of “maximal families of reals”, such as mad families,
ultrafilters, maximal indpendent families, Hamel bases or Vitali sets. Our main result
is that, after a countable support iteration of Sacks forcing or of splitting forcing (a less
known forcing adding splitting reals) over L, every analytic hypergraph on a Polish space
has a ∆12 maximal independent set. This means that in the models obtained by theseiterations, most types of interesting “maximal families” have ∆12 witnesses. As a corollary,
this solves an open problem of Brendle, Fischer and Khomskii by providing a model witha Π11 mif (maximal independent family) while the independence number i is bigger than
ℵ1 .

6Jaroslav Supina
Ideal Fréchet–Urysohn property of a space of continuous functions
Let us recall a folklore convergence concept by H. Cartan from 1937, stated in terms ofideals, for set of reals X and associated space Cp (X), the space of real-valued continuousfunctions on X with topology of pointwise convergence. For a continuous function f in
the closure of A ⊆ Cp (X) there is a sequence {fn : n ∈ ω} ⊆ A and an ideal I on natural
numbers such that ⟨fn : n ∈ ω⟩ converges to f with respect to I.
We shall focus on sets of reals X such that for a fixed ideal I and any continuous fin the closure of A ⊆ Cp (X) there is a sequence {fn : n ∈ ω} ⊆ A converging to f withrespect to I.
Some of the presented results were obtained with S. Bardyla, V. Šottová or L. Zdomskyy.

Yasuo Yoshinobu
Fragility of properness
We prove that for any models V ⊆ W of ZFC with the same ordinals, there is a poset
which is proper in V but not in W . This answers a question raised by Karagila.

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