## David Schrittesser: The Ramsey property, MAD families, and their multidimensional relatives

Talk held by David Schrittesser (KGRC) at the KGRC seminar on 2018-10-25.

Abstract: Suppose every set of real numbers has the Ramsey property and “uniformization on Ellentuck-comeager sets” as well as Dependent Choice hold (as is the case under the Axiom of Determinacy, but also in Solovay’s model). Then there are no MAD families. As it turns out, there are also no (Fin x Fin)-MAD families, where Fin x Fin is the two-dimensional Fubini product of the ideal of finite sets. We also comment on higher dimensional products.

All results are joint work with Asger Törnquist.

## Arturo Martínez-Celis: On the Michael Space Problem

Seminar: Working group in applications of set theory, IMPAN

Thursday, 25.10. 2018, 10:15, room 105, IMPAN

Speaker: Arturo Martínez-Celis (IM PAN)

Title: “On the Michael Space Problem”

Abstact: “A Lindelöf Topological space is Michael if it has non-Lindelöf product with the space of the irrational numbers. These kind of spaces were introduced by Ernest Michael in 1963 and it is still unknown if one can be constructed in ZFC. We will introduce the notion of Michael ultrafilter, which implies the existence of a Michael space. We will also discuss the relation between this kind of ultrafilters and some classical cardinal invariants and we will use this to study the behaviour of this notion in some models of set theory”.

Visit our seminar page which may include some future talks at https://www.impan.pl/~set_theory/Seminar/

## Frank Stephan: A fast exponential time algorithm for Max Hamming Distance X3SAT

Invitation to the Logic Seminar at the National University of Singapore

Date: Wednesday, 24 October 2018, 17:00 hrs

Room: S17#04-05, Department of Mathematics, NUS

Speaker: Frank Stephan

Title: A fast exponential time algorithm for Max Hamming Distance X3SAT

URL: http://www.comp.nus.edu.sg/~fstephan/logicseminar.html

SAT is the problem which asks whether a given list of clauses in
n Boolean variables can be satisfied by an assignment
to the variables which is common for all clauses,
XSAT is the variant of SAT which asks the same, but requires in
addition that in each clause exactly one literal is true; X3SAT
is the corresponding problem with the additional constraint that
every clause contains up to three literals only. Now the problem
“Max Hamming Distance X3SAT” asks for the maximum
Hamming distance taken by two solutions of a given instance.
X3SAT can be solved only in exponential time, but the corresponding
exponential time algorithm is quite fast, an algorithm of Magnus
Wahlstroem from 2007 provides time O(1.0984^n).
The problem to find pairs of solutions of maximum Hamming distance
is more difficult: The up to now best known bound is by
Fu, Zhou and Yin of time O(1.6760^n) and the contribution
of the here presented work is to bring this down to O(1.3298^n).

This is joint work with Gordon Hoi and the paper is available
from the speaker’s homepage.

## Stevo Todorcevic: A proof of Galvin’s Conjecture

Place: Fields Institute (Room 210)
Date: October 26, 2018 (13:30-15:00)
Speaker: Stevo Todorcevic
Title: A proof of Galvin’s Conjecture
Abstract: We prove that for every finite colouring of the set of unordered
pairs of real numbers there is a set of reals homeomorphic to the rationals whose pairs use no more than two colours. This solves a problem of F. Galvin from the 1970’s. The proof uses large cardinals. This is a joint work with Dilip Raghavan.

## Ilya Shapirovsky: Modal logics of model-theoretic relations

Place: Fields Institute (Room 210)
Date: October 19 , 2018 (13:30-15:00)
Speaker: Ilya Shapirovsky
Title: Modal logics of model-theoretic relations
Abstract: Consider a unary operation f on the set of sentences of a model-theoretic language L, and a set T of sentences of L.  Properties of f in T can be studied using  propositional modal language: variables are evaluated as sentences of L, and f interprets the modal operator. The modal theory of f in T is defined as the set of those modal formulas which are in T under every valuation.

An example of this approach is Solovay’s theorem providing a complete modal axiomatization of formal provability in Peano arithmetic. Another example is the theorem of Hamkins and Loewe axiomatizing the modal logic of forcing, where the modal operator expresses satisfiability in forcing extensions. Both these logics  have good semantic and algorithmic properties: in particular, they have the finite model property, are finitely axiomatizable, and hence decidable.

This raises the question of modal theories of other model-theoretic relations R (e.g., the submodel relation or the homomorphic image relation). These theories can be defined in the case when satisfiability in R-images is expressible in L. In this talk we will discuss general properties of such modal systems, and then provide a complete axiomatization for the case of the submodel relation.  This talk is based on a joint work with D.I. Saveliev.

## David Asperó: Special $\aleph_2$-Aronszajn trees and GCH

Talk held by David Asperó (University of East Anglia, Norwich, UK) at the KGRC seminar on 2018-10-22.

Abstract: In joint work with Mohammad Golshani, and assuming the existence of a weakly compact cardinal, we build a forcing extension in which GCH holds and every $\aleph_2$-Aronszajn tree is special. This answers a well-known question from the 1970’s. I will give the proof of this theorem, with as many details as possible.

## Fabiana Castiblanco: Capturing tree forcing notions and some preservation results

The seminar meets on Wednesday October 24th at 11:00 in the Institute of
Mathematics CAS, Zitna 25, seminar room, 3rd floor, front building.

Program: Fabiana Castiblanco — Capturing tree forcing notions and some
preservation results

In this talk, we will introduce the concept of capturing forcing notions
in order to show that various tree posets such as Sacks (S), Silver (V),
Mathias (M), Laver (L) and Miller (ML) forcing preserve the existence of
sharps for reals. Furthermore, these tree forcing notions preserve
levels of Projective Determinacy. As a consequence of this fact we
obtain that Σ^1_{n+3}-P-absoluteness holds for P∈T := {S, V, M, L, ML}
under the assumption of Π^1_{n+1}-determinacy.
If time permits, as an application of our results, we will see that if
Π^1_{n+1}-determinacy holds, each P∈T does not add new orbits to
∆^1_{n+3}-thin transitive relations.

Best,
David

## Dilip Raghavan: Order dimension

Invitation to the Logic Seminar at the National University of Singapore

Date: Wednesday, 17 October 2018, 17:00 hrs

Room: S17#04-05, Department of Mathematics, NUS

Speaker: Dilip Raghavan

Title: Order dimension

URL: http://www.comp.nus.edu.sg/~fstephan/logicseminar.html

Abstract: We will present some results on the order dimension of
various partial orders, focusing on partial orders which are locally countable.
This is joint work with Kojiro Higuchi, Steffen Lempp and Frank Stephan.

## Reflections on Set Theoretic Reflection, Sant Bernat, November 16-19, 2018

Reflections on Set Theoretic Reflection

16-19 November 2018

Sant Bernat, Montseny, Catalonia

Set Theory Conference

Organised in partnership with the Clay Mathematics Institute

Keynote Speakers

• Peter Koellner (Harvard)
• Paul Larson (Miami)
• Menachem Magidor (Jerusalem)
• Jouko Väänänen (Helsinki and Amsterdam)
• W. Hugh Woodin (Harvard)

Organisers

• David Asperó (UEA, Norwich)
• Carles Casacuberta (IMUB, Barcelona)
• Miguel Ángel Mota (ITAM, México)
• Konstantinos Tsaprounis (UAegean, Samos)

Venue and Accommodation

The event will take place at Hotel Sant Bernat, located in the Montseny natural park, in Catalonia.

Lodging rates are in Euros and include three hotel nights with three meals per day (from lunch on Friday 16 up to breakfast on Monday 19) plus tourist taxes.

• Double room for single use: 427 €
• Shared double room: 317 €
• Shared triple room: 295 €

There is the possibility of arriving earlier and/or departing later from the hotel. In such a case, the additional costs are as follows (room prices refer to one extra night, including breakfast):

• Double room for single use: 70 €
• Shared double room: 43 €
• Lunch on Monday: 26 €

Transportation

Both on Friday 16 (arrival) and on Monday 19 (departure), transportation between El Prat airport of Barcelona and the hotel, passing through downtown Barcelona, will be arranged by the organisers. The bus ride from the airport to the hotel can take around two hours.

Further information on location and transportation is given here, including a map, bus schedule, car ride directions, train and taxi options.

Registration

The registration fee amounts to 75 € and includes a welcome cocktail, coffee breaks, and a mushroom hunting snack on Saturday 17. Instructions for registration and payment will be sent to pre-registered participants. Please contact the organisers for pre-registration.

List of Participants

So far the following people have pre-registered for the conference.

Contact

You may contact the organisers either directly or via the e-mail address 2018bagaria60 AT gmail DOT com.

The organisers gratefully acknowledge support from the Clay Mathematics Institute, the Foundation Compositio Mathematica, and the Institute of Mathematics of the University of Barcelona.

## Anush Tserunyan: Hyperfinite subequivalence relations of treed equivalence relations

Mathematical logic seminar – Oct 16 2018
Time:     3:30pm – 4:30 pm

Room:     Wean Hall 8220

Speaker:         Anush Tserunyan
Department of Mathematics
University of Illinois at Urbana-Champaign

Title:     Hyperfinite subequivalence relations of treed equivalence relations

Abstract:

A large part of measured group theory studies structural properties of countable groups that hold “on average”. This is made precise by studying the orbit equivalence relations induced by free Borel actions of these groups on probability spaces. In this vein, the cyclic (more generally, amenable) groups correspond to hyperfinite equivalence relations, and the free groups to the treeable ones. In joint work with R. Tucker-Drob, we give a detailed analysis of the structure of hyperfinite subequivalence relations of a treed equivalence relation, deriving some of analogues of structural properties of cyclic subgroups of a free group. In particular, just like any cyclic subgroup is contained in a unique maximal one, we show that any hyperfinite subequivalence relation is contained in a unique maximal one.