A strong minimal pair in r.e. degrees is defined to be a pair of
A, B such that they are incomparable and for any non-recursive r.e. set W
below A, B+W computes A. Historically, this was a difficult problem.
Slaman showed a weaker version of this (i.e. B+W computes a third set C,
instead of A), and it is called Slaman-Triple nowadays. Only recently,
people showed that there is a strong minimal pair. However, we realized
that there is a problem in that paper. Then we turned the problem into a
proof that there is no strong minimal pair. In this talk, we will sketch
The seminar meets on Wednesday February 20th at 11:00 in the Institute
of Mathematics CAS, Zitna 25, seminar room, 3rd floor, front building.
Program: Asaf Karagila — Staring into a Cohen real: the Bristol model
What kind of intermediate models can be found when adding a Cohen real,
c, to L? If we are concerned with models of ZFC, then answer is simple:
L itself, or Cohen extensions of L.
But when models of ZF are of interest, this fails spectacularly. The
Bristol model is a model intermediate to L[c] which is not even
constructible from a set. We will discuss the details of the
construction, and the consequences it has on the models which are
trapped between L and L[c].
Place: Fields Institute (Room 210)
Date: February 15, 2018 (13:30-15:00)
Speaker: David Schrittesser
Title: The Ramsey property, MAD families, and their multidimensional relatives
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. Results are joint work with Asger Törnquist.
Set-theoretic methods in topology and real functions theory,
dedicated to 80th birthday of Lev Bukovsky
September 9-13, 2019
The conference takes place in Kosice, the city located in eastern Slovakia. There is a nice occasion of celebrating 80th birthday of Lev Bukovsky this year, who spent most of his life in the city. The webpage of the meeting is
Topics of interest include set-theoretic topology, descriptive set theory, cardinal invariants, selection principles and set-theoretic aspects of the convergence of functions.
The program consists of lectures provided by invited speakers:
and contributed talks of participants.
Participants registration with welcome reception starts on Sunday evening, September 8. We plan to open the online registration in March at the webpage of the meeting. Due to Visegrad Fund we may partially support the accommodation of small number of PhD students or young researchers, from Visegrad Countries, Eastern Partnership Countries or Western Balkans Countries.
The seminar meets on Wednesday February 13th at 11:00 in the Institute of Mathematics CAS, Zitna 25, seminar room, 3rd floor, front building.
Program: Jan Hubička — Combinatorial proofs of the extension property for partial automorphisms
Class K of finite structures has extension property for partial automorphisms (EPPA) if for every A in K there exists B in K such that every partial automorphism of A (that is isomorphism of two substructures of A) extends to automorphism of B. Hrushovski, in 1992,
shown that the class of all finite graphs has EPPA. This result was used by Hodges, Hodkinson, Lascar and Shelah to show that the random
graph has small index property. This motivated search for new classes with EPPA. I will show (and partly prove) new general theorem giving a structural condition for class having EPPA. The theorem is a strengthening of the Herwig–Lascar theorem, but the proof techniques are new, combinatorial and completely self-contained.
I will also discuss connection to structural Ramsey theory.
This is joint work with Jaroslav Nesetril and Matej Konecny.
We give examples of analytic sequent calculi LK+ and LK++ that
extend Gentzen’s sequent calculus LK by unsound quantifier rules
in such a way that (i) derivations lead only to true sequents
(ii) cut free proofs may be non-elementary shorter than cut free LK proofs.
This research is based on properties of Hilbert’s epsilon calculus and
is part of efforts to complement Hilber’s stepwise concept of proof by
useful global concepts.
We use these ideas to provide analytic calculi for Henkin quantifiers and
demonstrate soundness, (cut free) completeness and cut elimination.
Furthermore, we show, that in the case of quantifier macros such as Henkin
quantifiers for a partial semantics global calculi are the only option to
Sierpinski’s now classical result states that there is an edge 2-colouring of the complete graph on aleph1 vertices so that there are no uncountable monochromatic subgraphs. In the 1970s, Erdos, Galvin and Hajnal asked what other graphs with large chromatic number admit similar edge colourings i.e., with no ‘large’ monochromatic subgraphs. We plan to review some recent advances on this problem and in particular, connect the question to Shelah’s ladder system uniformization theory.
Speaker: Clinton Conley
Department of Mathematical Sciences
Title: Ode on a one-ended subforest
Many arguments in (finite) graph theory follow this pattern: postpone some
onerous task until the last possible moment, after you’ve arranged things
to make the task as easy as possible. In the descriptive set-theoretic
milieu, the one-ended forest provides a portal to a procrastinator’s
wonderland in which the onerous task instead wanders off to infinity. We
discuss a few instances of this phenomenon and some applications to
coloring and treeing graphs.
BARCELONA RESEARCH GROUP IN SET THEORY
BARCELONA SET THEORY SEMINAR
An invitation to the world of Prikry-type forcing
Universitat de Barcelona
Abstract: Prikry-type forcing plays a central role in
combinatorics due to its close links with central set-theoretic
principles such as the Singular Cardinal Hypothesis (SCH) or the
Tree Property. The original Prikry forcing was devised to change
the cofinality of a measurable cardinal to ω, but there are now
many other more sophisticated constructions that yield more
powerful applications. Among them we can find Magidor and
Radin forcing for changing cofinalities to uncountable cardinals,
or the Diagonal supercompact Prikry forcing with collapses, due
also to Magidor, which can be used to force the failure of SCH
at the first singular cardinal. In this session we will give an
introduction from the very beginning to this family of forcings,
and if time permits we will present some easy applications.
Date: Thursday 14 February 2019
Place: Room S-1*
Departament de Matemàtiques i Informàtica
Universitat de Barcelona
Gran Via de les Corts Catalanes 585, 08007 Barcelona
* Enter the University building through the door 20 meters to the right of the
main door and, as you enter the courtyard, turn left, go to the end of the
corridor, and then downstairs.
Place: Fields Institute (Room 210)
Date: February 8 , 2019 (13:30-15:00)
Speaker: Damjan Kalajdzievski
Title: How to show Con(ZFC + omega_1=u<a) from Con(ZFC)
Abstract: I will outline how to prove the result in the title by joint work with Osvaldo Guzman