## Nicholas Ramsey: NSOP_1 Theories

BGU seminar in Logic, Set Theory and Topology.

Time: Tuesday, April 25th, 12:15-13:30.
Place: Seminar room -101, Math building 58.
Speaker: Nicholas Ramsey (UC Berkeley)

Title: NSOP_1 Theories

Abstract:
The class of NSOP_1 theories was isolated by Džamonja and Shelah in the mid-90s and later investigated by Shelah and Usvyatsov, but the theorems about this class were mainly restricted to its syntactic properties and the model-theoretic general consensus was that the property SOP_1 was more of an unimportant curiosity than a meaningful dividing line. I’ll describe recent work with Itay Kaplan which upends this view, characterizing NSOP_1 theories in terms of an independence relation called Kim-independence, which generalizes non-forking independence in simple theories.  I’ll describe the basic theory and describe several examples of non-simple NSOP_1 theories, such as Frobenius fields and vector spaces with a generic bilinear form.

## Thilo Weinert: Partitioning pairs of sigma-scattered linear orders

BIU seminar in Set Theory

On 20/04/2017, 10-12, Building 604, Room 103

Speaker:  Thilo Weinert

Title: Partitioning pairs of sigma-scattered linear orders

Abstract. We are going to continue the analysis of generalised scattered orders, proving the result described towards the end of Chris Lambie-Hanson’s talk. This states that consistently, for every sigma-scattered linear ordering there is a colouring of its pairs in black & white such that every triple contains a white pair and every copy of one of the following order-types contains a black pair:

• omega_1^omega
• (omega_1^omega)^*
• omega_1 * (omega * omega^*)^omega
• omega_1^* * (omega * omega^*)^omega
• (omega * omega^*)^omega * omega_1
• (omega * omega^*)^omega * omega_1^*

This generalises a 46-year-old Theorem of Erdős & Rado about ordinals. A sufficient hypothesis implying this theorem is the existence of a colouring of the pairs of omega_1 * omega in black & white such that every triple contains a black pair and every subset of full order-type contains a white one. Time permitting we may present a proof that stick = b = Aleph_1 implies the existence of such a colouring. Here b is the unbounding number and stick = Aleph_1 is a weakening of the club principle which was considered by Baumgartner 41 years ago, named by Broverman, Ginsburg, Kunen & Tall two years thereafter and twenty years ago reconsidered as a cardinal characteristic by Fuchino, Shelah & Soukup.

## Assaf Rinot: Distributive Aronszajn trees

Forcing Seminar (Tel-Aviv University)

Tuesday, 4/Apr/2017, 9-11.
Room 007, Schriber building, Tel-Aviv University.

Speaker: Assaf Rinot

Title: Distributive Aronszajn trees

Abstract: We address a conjecture asserting that, assuming GCH, for every singular cardinal $\lambda$, if there exists a $\lambda^+$-Aronszajn tree, then there exists one which is moreover $\lambda$-distributive.

## Omer Mermelstein: Closed ordinal Ramsey numbers below $\omega^\omega$

BGU seminar in Logic, Set Theory and Topology
Time: Tuesday, April 4th, 12:15-13:30.
Place: Seminar room -101, Math building 58.
Speaker: Omer Mermelstein (BGU)

Title: Closed ordinal Ramsey numbers below $\omega^\omega$

Abstract:Since the 1950s, many versions of the partition calculus and arrow notation, introduced by Erdős and Rado, were studied. One such variant, introduced by Baumgartner and recently studied by Caicedo and Hilton, is the closed ordinal Ramsey number. For this variant, we require our homogeneous subset to be both order-isomorphic and homeomorphic to a given ordinal.

In the talk we present an approach with which to tackle this flavour of partition calculus, and if time permits prove some results. The talk is elementary and self-contained.

## Chris Lambie-Hanson: Partition relations and generalized scattered orders

BIU seminar in Set Theory

On 30/03/2017, 10-12, Building 604, Room 103

Speaker: Chris Lambie-Hanson

Title: Partition relations and generalized scattered orders
Abstract: The class of scattered linear orders, isolated by Hausdorff, plays a prominent role in the study of general linear orders. In 2006, Dzamonja and Thompson introduced classes of orders generalizing the class of scattered orders. For a regular cardinal kappa, they defined the classes of kappa-scattered and weakly kappa-scattered linear orders. For kappa = omega, these two classes coincide and are equal to the classical class of scattered orders. For larger values of kappa, though, the two classes are provably different. In this talk, we will investigate properties of these generalized scattered orders with respect to partition relations, in particular the extent to which the classes of kappa-scattered or weakly kappa-scattered linear orders of size kappa are closed under partition relations of the form tau -> (phi, n) for n < omega. We will show that, assuming kappa^{<kappa} = kappa, the class of weakly kappa-scattered orders is closed under all such partition relations while, for uncountable values of kappa, the class of kappa-scattered orders consistently fails to be closed. Along the way, we will prove a generalization of the Milner-Rado paradox and look at some results regarding ordinal partition relations. This is joint work with Thilo Weinert.

## Chris Lambie-Hanson: Trees with ascent paths

HUJI Logic Seminar

The next meeting of the Logic Seminar will be in Wednesday, 22/03/17, 16:00 – 18:00, Ross buiding.

Speaker: Chris Lambie-Hanson

Title: Reflections on the coloring and chromatic numbers

Abstract: The notion of an ascent path through a tree, isolated by Laver, is a generalization of the notion of a cofinal branch and, in many cases, the existence of an ascent path through a tree provides a concrete obstruction to the tree being special. We will discuss some recent results regarding ascent paths through $\kappa$-trees, where $\kappa > \omega_1$ is a regular cardinal. We will discuss the consistency of the existence or non-existence of a special $\mu^+$-tree with a $cf(\mu)$-ascent path, where $\mu$ is a singular cardinal. We will also discuss the consistency of the statement, “There are $\omega_2$-Aronszajn trees but every $\omega_2$-tree contains an $\omega$-ascent path.” We will connect these topics with various square principles and with results about the productivity of chain conditions.

## Chris Lambie-Hanson: Reflections on the coloring and chromatic numbers

HUJI Logic Seminar

The next meeting of the Logic Seminar will be in Wednesday, 18/01, 16:00 – 18:00, Ross 70.
Speaker: Chris Lambie-Hanson

Title: Reflections on the coloring and chromatic numbers

Abstract: Compactness phenomena play a central role in modern set theory, and the investigation of compactness and incompactness for the coloring and chromatic numbers of graphs has been a thriving area of research since the mid-20th century,when De Bruijn and Erdős published their compactness theorem for finite chromatic numbers.

In this talk, we will briefly review some of the highlights in this area and then present new results indicating, firstly, that the coloring number can only exhibit a limit amount of incompactness, and, secondly, that large amounts of incompactness for the chromatic number are compatible with strong compactness statements, including compactness for the coloring number.
This is joint work with Assaf Rinot.

## Salma Kuhlmann: The Baer-Krull Theorem for Quasi-ordered fields

BGU Seminar in Logic, Set Theory and Topology

Tomorrow we continue our seminar in Logic, Set Theory and Topology.
Time: Tuesday, January 3, 12:15-13:30.
Place: Seminar room -101, Math building 58.
Speaker: Salma Kuhlmann (Konstantz)
Title: The Baer-Krull Theorem for Quasi-ordered fields
Abstract:
In my seminar talk on 29.12.2015, I introduced the notion of quasi-ordered fields, proved Fakhruddin’s dichotomy. In this talk, I will present a version of a classical theorem in real algebra (the Baer-Krull theorem) for quasi-ordered fields.

## Ludomir Newelski: Stable groups and topological dynamics

HUJI Logic Seminar

The next meeting of the Logic Seminar will be in Wednesday, 04/01/17, between 16:00 – 18:00, Ross 70.

Please forward this mail to anyone that might be interested.
Speaker: Ludomir Newelski
Title: Stable groups and topological dynamics

Abstract: Assume G is a stable group. I will recall an old 2-step theorem of mine on generating a type-definable subgroup of G by a single type. I will discuss some related questions and put them into context of topological dynamics.

## Itaï Ben Yaacov: Baby version of the asymptotic volume estimate

HUJI Logic Seminar

Tomorrow (27/12) we will have a lecture of Itaï Ben Yaacov in Sprinzak Building, between 10:00 – 12:00, room 102.

Please forward this mail to anyone that might be interested.

Baby version of the asymptotic volume estimate

Abstract: I’ll show how the Vandermonde determinant identity allows us to

estimate the volume of certain spaces of polynomials in one variable

(or rather, of homogeneous polynomials in two variables), as the degree

goes to infinity.

I’ll explain what this is good for in the context of globally valued

fields, and, given time constraints, may give some indications on the

approach for the “real inequality” in higher projective dimension.