## Chris Lambie-Hanson: A forcing axiom deciding the generalized Souslin Hypothesis

Mathematical logic seminar – Oct 3 2017
Time:     3:30pm – 4:30 pm

Room:     Wean Hall 8220

Speaker:         Chris Lambie-Hanson
Department of Mathematics
Bar-Ilan University

Title:     A forcing axiom deciding the generalized Souslin Hypothesis

Abstract:

Given a regular, uncountable cardinal $\kappa$, it is often desirable to be able to construct objects of size $\kappa^+$ using approximations of size less than $\kappa$. Historically, such constructions have often been carried out with the help of a $(\kappa,1)$-morass and/or a $\diamondsuit(\kappa)$-sequence.
We present a framework for carrying out such constructions using $\diamondsuit(\kappa)$ and a weakening of Jensen’s $\square_\kappa$. Our framework takes the form of a forcing axiom, $\textrm{SDFA}(\mathcal P_\kappa)$. We show that $\textrm{SDFA}(\mathcal P_κ)$ follows from the conjunction of $\diamondsuit(\kappa)$ and our weakening of $\square_\kappa$ and, if $\kappa$ is the successor of an uncountable cardinal, that $\textrm{SDFA}(\mathcal P_\kappa)$ is in fact equivalent to this conjunction. We also show that, for an infinite cardinal $\lambda$, $\textrm{SDFA}(\mathcal P_{\lambda^+})$ implies the existence of a $\lambda^+$-complete $\lambda^{++}$-Souslin tree. This implies that, if $\lambda$ is an uncountable cardinal, $2^\lambda =\lambda^+$, and Souslin’s Hypothesis holds at $\lambda^{++}$, then $\lambda^{++}$ is a Mahlo cardinal in $L$, improving upon an old result of Shelah and Stanley. This is joint work with Assaf Rinot.

## Garrett Ervin: The Cube Problem for linear orders II

Mathematical logic seminar – Sep 26 2017
Time:     3:30pm – 4:30 pm

Room:     Wean Hall 8220

Speaker:         Garrett Ervin
Department of Mathematical Sciences
CMU

Title:     The Cube Problem for linear orders II

Abstract:

In the 1950s, Sierpiński asked whether there exists a linear order that is isomorphic to its lexicographically ordered Cartesian cube but not to its square. The analogous question has been answered positively for many different classes of structures, including groups, Boolean algebras, topological spaces, graphs, partial orders, and Banach spaces. However, the answer to Sierpinski’s question turns out to be negative: any linear order that is isomorphic to its cube is already isomorphic to its square, and thus to all of its finite powers. I will present an outline of the proof and give some related results.

## Garrett Ervin: The Cube Problem for linear orders

Mathematical logic seminar – Sep 19 2017
Time:     3:30pm – 4:30 pm

Room:     Wean Hall 8220

Speaker:         Garrett Ervin
Department of Mathematical Sciences
CMU

Title:     The Cube Problem for linear orders

Abstract:

In the 1950s, Sierpiński asked whether there exists a linear order that is isomorphic to its lexicographically ordered Cartesian cube but not to its square. The analogous question has been answered positively for many different classes of structures, including groups, Boolean algebras, topological spaces, graphs, partial orders, and Banach spaces. However, the answer to Sierpinski’s question turns out to be negative: any linear order that is isomorphic to its cube is already isomorphic to its square, and thus to all of its finite powers. I will present an outline of the proof and give some related results.

## Marcos Mazari Armida: Introduction to good frames in Abstract Elementary Classes

Hello,

The seminar will continue to meet on Mondays in WeH 8201 at 5PM, the talks usually last 90 minutes.
Marcos Mazari Armida will give at least three talks, introducing Shelah’s good frames which the generalization to Abstract Elementary Classes of forking, he will focus on obtaining exists theorem of models when model theoretic assumptions will be replacing rather article non-ZFC axioms used by Shelah.
Information on this seminar is posted on the departmental web page http://www.math.cmu.edu/math/modeltheoryseminars/modeltheoryseminar.php?SeminarSelect=1548  or see below.
Best,
Rami Grossberg.
——————————————————-

Marcos Mazari Armida

Carnegie Melllon University
Title: Introduction to good frames in Abstract Elementary Classes, Part 1

Abstract:
The central notion of Shelah’s book on Abstract Elementary Classes [Sh:h] is the notion of a good $\lambda$-frame. It is a forking like notion for types over models of size $\lambda$ and the existence of it implies that the class is well-behaved in $\lambda$. In this series of talks we will focus on the question of the existence of extensions to models of size greater than $\lambda$. We will prove that under some reasonable hypothesis it is always possible to extend a frame. One interesting corollary of this is the existence of arbitrary large models, this is done within ZFC. The first couple of lectures will be based on [Sh:h] Chapter II, while our main theorem is the main theorem of [Bon14a].

References:
[Bon14a] Will Boney, Tameness and extending frames, Journal of Mathematical Logic 14, no. 2
[Sh:h] Saharon Shelah, Classication Theory for Abstract Elementary Classes, vol. 1 & 2, Mathematical Logic and Foundations, no. 18 & 20, College Publications, 2009.

Date: Monday , September 18, 2017.
Time: 5:00 pm
Location: Wean Hall 8201

Marcos Mazari Armida

Carnegie Melllon University
Title: Introduction to good frames in Abstract Elementary Classes, Part 2

Abstract:
The central notion of Shelah’s book on Abstract Elementary Classes [Sh:h] is the notion of a good $\lambda$-frame. It is a forking like notion for types over models of size $\lambda$ and the existence of it implies that the class is well-behaved in $\lambda$. In this series of talks we will focus on the question of the existence of extensions to models of size greater than $\lambda$. We will prove that under some reasonable hypothesis it is always possible to extend a frame. One interesting corollary of this is the existence of arbitrary large models, this is done within ZFC. The first couple of lectures will be based on [Sh:h] Chapter II, while our main theorem is the main theorem of [Bon14a].

References:
[Bon14a] Will Boney, Tameness and extending frames, Journal of Mathematical Logic 14, no. 2
[Sh:h] Saharon Shelah, Classication Theory for Abstract Elementary Classes, vol. 1 & 2, Mathematical Logic and Foundations, no. 18 & 20, College Publications, 2009.

Date: Monday , September 25, 2017.
Time: 5:00 pm
Location: Wean Hall 8201

Marcos Mazari Armida

Carnegie Melllon University
Title: Introduction to good frames in Abstract Elementary Classes, Part 3

Abstract:
The central notion of Shelah’s book on Abstract Elementary Classes [Sh:h] is the notion of a good $\lambda$-frame. It is a forking like notion for types over models of size $\lambda$ and the existence of it implies that the class is well-behaved in $\lambda$. In this series of talks we will focus on the question of the existence of extensions to models of size greater than $\lambda$. We will prove that under some reasonable hypothesis it is always possible to extend a frame. One interesting corollary of this is the existence of arbitrary large models, this is done within ZFC. The first couple of lectures will be based on [Sh:h] Chapter II, while our main theorem is the main theorem of [Bon14a].

References:
[Bon14a] Will Boney, Tameness and extending frames, Journal of Mathematical Logic 14, no. 2
[Sh:h] Saharon Shelah, Classication Theory for Abstract Elementary Classes, vol. 1 & 2, Mathematical Logic and Foundations, no. 18 & 20, College Publications, 2009.

Date: Monday , October 2, 2017.
Time: 5:00 pm
Location: Wean Hall 8201

## Andy Zucker: A direct solution to the Generic Point Problem II

Mathematical logic seminar – Sep 12 2017
Time:     3:30pm – 4:30 pm

Room:     Wean Hall 8220

Speaker:         Andy Zucker
Department of Mathematical Sciences
CMU

Title:     A direct solution to the Generic Point Problem II

Abstract:

We provide a new proof of a recent theorem of Ben-Yaacov, Melleray, and Tsankov. If G is a Polish group and X is a minimal, metrizable G-flow with all orbits meager, then the universal minimal flow M(G) is non-metrizable. In particular, we show that given X as above, the universal highly proximal extension of X is non-metrizable.

## Andy Zucker: A direct solution to the Generic Point Problem

Mathematical logic seminar – Sep 5 2017
Time:     3:30pm – 4:30 pm

Room:     Wean Hall 8220

Speaker:         Andy Zucker
Department of Mathematical Sciences
CMU

Title:     A direct solution to the Generic Point Problem

Abstract:

We provide a new proof of a recent theorem of Ben-Yaacov, Melleray, and Tsankov. If G is a Polish group and X is a minimal, metrizable G-flow with all orbits meager, then the universal minimal flow M(G) is non-metrizable. In particular, we show that given X as above, the universal highly proximal extension of X is non-metrizable.

## Natasha Dobrinen: The big Ramsey degrees for the universal triangle-free graph

Mathematical logic seminar – Jun 5 2017
Time:     3:00pm – 4:00 pm

Room:     Wean Hall 8220

Speaker:         Natasha Dobrinen
Department of Mathematics
University of Denver

Title:     The big Ramsey degrees for the universal triangle-free graph

Abstract:

The Rado graph (aka the countable random graph) is the unique countable graph G which is:

a) Universal, that is G contains an induced copy of every finite graph.

b) Homogeneous, that is any isomorphism between finite induced subgraphs
of G extends to an automorphism of G.

The construction of the Rado graph works with many classes of finite structures (the Fraïssé classes), assigning to each Fraïssé class a countable, universal and homogeneous structure called the Fraïssé limit.

Ramsey theory on relational structures can be studied from two vantage points. The first, more classical, is to study when, given two finite structures A and B and given any k greater than 1, there is another finite structure C such that for any coloring of all copies of A in C into k colors, there is a copy of B in C in which all copies of A have the same color. A Fraïssé class of finite relational structures has the Ramsey property if this holds for any two structures A and B in the class. Nešetřil and Rödl have shown that many classes of finite ordered relational structures have the Ramsey property, including finite ordered graphs and finite ordered triangle-free graphs.

The second, and of much recent interest, is to study colorings of copies of a finite structure inside an infinite homogenous structure, usually the Fraïssé limit of some Fraïssé class of finite structures. It has been shown that any finite coloring of the vertices of the Rado graph can be reduced to one color on a subgraph which is also a Rado graph. For edges and other structures with more than one vertex, Sauer has proved this to be impossible. However, he also proved that given a finite graph A, there is a number n(A) such that any coloring of all copies of A in the Rado graph into finitely many colors may be reduced to n(A) colors on a copy of the Rado graph. We say, then, that the Rado graph has finite big Ramsey degrees. Similar results have been obtained for other countable homogeneous structures, though many are still open.

We have looked at the problem of finite big Ramsey degrees for the universal triangle-free graph H, that is, the homogeneous graph with no triangles into which every countable triangle-free graph embeds. This is the first homogeneous structure omitting a subtype to be addressed for big Ramsey degrees. Using the method of forcing, but in ZFC, we prove a new Ramsey theorem on trees which code H, and apply it to deduce that H has finite big Ramsey degrees.

## James Cummings: Definable subsets of singular cardinals

Mathematical logic seminar – Apr 11 2017
Time:     3:30pm – 4:30 pm

Room:     Wean Hall 8220

Speaker:         James Cummings
Department of Mathematical Sciences
CMU

Title:     Definable subsets of singular cardinals

Abstract:

Shelah proved the surprising result that if μ is a singular strong limit cardinal of uncountable cofinality, then there is a subset X of μ such that all subsets of μ are ordinal-definable from X. We will give a proof and discuss some complementary consistency results.

## Jing Zhang: A polarized partition theorem for large saturated linear orders

Mathematical logic seminar – Mar 28 2017
Time:     3:30pm – 4:30 pm

Room:     Wean Hall 8220

Speaker:         Jing Zhang
Department of Mathematical Sciences
CMU

Title:     A polarized partition theorem for large saturated linear orders

Abstract:     Laver proved the following polarized partition theorem for rational numbers: for any natural number d, any finite coloring f of Q^d, there exist subsets of Q, X_i for i < d, each of which has the same order type as Q such that the product X_1 x … x X_{d-1} gets at most d! many colors. A natural question to ask is what happens when we consider larger saturated linear orders. We will discuss the consistency at the level of strongly inaccessible cardinals that satisfy some indestructibility property. The development of versions of the Halpern-Läuchli theorem at a large cardinal will be pivotal in the proof.

## Dana Bartošová: Freedom of action in combinatorial terms

Mathematical logic seminar – Mar 21 2017
Time:     3:30pm – 4:30 pm

Room:     Wean Hall 8220

Speaker:         Dana Bartošová
Department of Mathematical Sciences
CMU

Title:     Freedom of action in combinatorial terms

Abstract:

A group acts freely on a compact Hausdorff space if all of its non-identity elements act without fixed points. By Veech’s theorem, every locally compact topological group admits a free action and the question arises to which other groups this property can be extended. On the other hand, elements of extremely amenable groups act with fixed points under any action but the opposite implication does not hold. We show a combinatorial reformulation of this property and ask how far it is from extreme amenability.