Archives of: Michigan Logic Seminar

Andres Caicedo: Preserving sequences of stationary subsets of omega_1

Thursday, November 10, 2016, from 4 to 5:30pm
East Hall, room 3096

Speaker: Andres Caicedo (Math Reviews)

Title: Preserving sequences of stationary subsets of omega_1

Abstract:

Let M be an inner model that computes omega_1 correctly. We show two results (due to Stevo Todorcevic and Paul Larson) on whether there is in M a partition of omega_1 into infinitely many sets that are stationary from the point of view of V.

David Fernández Bretón: Strong failures of higher analogs of Hindman’s theorem, IV

Thursday, November 3, 2016, from 4 to 5:30pm
East Hall, room 3096

Speaker: David Fernández Bretón (University of Michigan)

Title: Strong failures of higher analogs of Hindman’s theorem, IV

Abstract:

This is talk 4 out of 4. Last time we introduced a set-theoretic principle S(kappa,theta), which implies a particular anti-Ramsey theoretic result on all groups of cardinality kappa. In this talk we will prove that S(omega_1,omega_1) holds, and also that S(kappa,omega) holds whenever kappa has uncountable cofinality equals the cofinality of the powerset of a cardinal lambda such that lambda^{<lambda}=lambda (this last result has a nice implication for the additive group of real numbers).

David J. Fernández Bretón: Strong failures of higher analogs of Hindman’s theorem, III

Thursday, October 27, 2016, from 4 to 5:30pm
East Hall, room 3096

Speaker: David J. Fernández Bretón (University of Michigan)

Title: Strong failures of higher analogs of Hindman’s theorem, III

Abstract:

Assuming the Continuum Hypothesis, Hindman, Leader and Strauss recently exhibited a colouring of the real line with two colours such that, for every uncountable set of reals, the collection of pairwise sums of these reals is panchromatic. We will show a few generalizations of these results, obtaining colourings both of the real line, and of other abelian groups, in many colours, satisfying similar anti-Ramsey-theoretic properties. This is talk number 3 out of n (where n is still TBD), and its contents are joint work with Assaf Rinot.

David J. Fernández Bretón: Strong failures of higher analogs of Hindman’s theorem, II

Thursday, October 20, 2016, from 4 to 5:30pm
East Hall, room 3096

Speaker: David J. Fernández Bretón (University of Michigan)

Title: Strong failures of higher analogs of Hindman’s theorem, II

Abstract:

(One of the versions of) Hindman’s theorem states that, whenever we partition an infinite abelian group G in two cells, there exists an infinite subset X of G such that the set FS(X) consisting of all sums of finitely many distinct elements of X is entirely contained within one of the cells of the partition. In this talk we will show that, when one attempts to replace both instances of “infinite” with “uncountable” in the theorem above, the resulting statement is not only false, but actually very false. This is talk 2 out of n (where n is a still unknown countable ordinal greater than or equal to 2). Joint work with Assaf Rinot.

David J. Fernández Bretón: Strong failures of higher analogs of Hindman’s theorem, I

Thursday, October 13, 2016, from 4 to 5:30pm
East Hall, room 3096

Speaker: David J. Fernández Bretón (University of Michigan)

Title: Strong failures of higher analogs of Hindman’s theorem, I

Abstract:

(One of the versions of) Hindman’s theorem states that, whenever we partition an infinite abelian group G in two cells, there exists an infinite subset X of G such that the set FS(X) consisting of all sums of finitely many distinct elements of X is entirely contained within one of the cells of the partition. In this talk we will show that, when one attempts to replace both instances of “infinite” with “uncountable” in the theorem above, the resulting statement is not only false, but actually very false. This is talk 1 out of n (where n is a still unknown nonzero countable ordinal). Joint work with Assaf Rinot.

Scott Schneider: Commuting endomorphisms and hypersmooth equivalence relations, III

Thursday, October 6, 2016, from 4 to 5:30pm
East Hall, room 3096

Speaker: Scott Schneider (University of Michigan)

Title: Commuting endomorphisms and hypersmooth equivalence relations, III

Abstract:

An equivalence relation E is hypersmooth (hyperfinite) if E is the union of an increasing sequence of smooth (finite) Borel equivalence relations. In the mid 80s, Weiss proved that the equivalence relation generated by a finite family of commuting Borel automorphisms is hyperfinite, and in the mid 90s, Dougherty, Jackson, and Kechris proved that the equivalence relation generated by a single Borel endomorphism is hypersmooth. We will generalize both results to show that the equivalence relation generated by a finite family of commuting Borel endomorphisms is hypersmooth. As is typical in this area, the proof will involve the construction of a suitable family of Borel marker sets. This is the third, and last, of this series of talks.

Scott Schneider: Commuting endomorphisms and hypersmooth equivalence relations, II

Thursday, September 29, 4:00-5:30, East Hall 3096.

An equivalence relation E is hypersmooth (hyperfinite) if E is the union of an increasing sequence of smooth (finite) Borel equivalence relations.  In the mid 80s, Weiss proved that the equivalence relation generated by a finite family of commuting Borel automorphisms is hyperfinite, and in the mid 90s, Dougherty, Jackson, and Kechris proved that the equivalence relation generated by a single Borel endomorphism is hypersmooth.  We will generalize both results to show that the equivalence relation generated by a finite family of commuting Borel endomorphisms is hypersmooth.  As is typical in this area, the proof will involve the construction of a suitable family of Borel marker sets.  This is the second of a series of talks.

Scott Schneider: Commuting endomorphisms and hypersmooth equivalence relations

Thursday, September 22, 4:00-5:30, East Hall 3096.

An equivalence relation E is hypersmooth (hyperfinite) if E is the union of an increasing sequence of smooth (finite) Borel equivalence relations.  In the mid 80s, Weiss proved that the equivalence relation generated by a finite family of commuting Borel automorphisms is hyperfinite, and in the mid 90s, Dougherty, Jackson, and Kechris proved that the equivalence relation generated by a single Borel endomorphism is hypersmooth.  We will generalize both results to show that the equivalence relation generated by a finite family of commuting Borel endomorphisms is hypersmooth.  As is typical in this area, the proof will involve the construction of a suitable family of Borel marker sets.  This talk will be part 1 of 2.

Ioannis Souldatos: The Hanf number for Scott sentences of computable structures.

Thursday, April 21, 2016, 4:00–5:30 PM, 3088 East Hall.

We will prove the following two theorems:
Theorem 1: Let A be a computable structure for a computable vocabulary \tau, and let \sigma be a Scott sentence for A. If \sigma has models of cardinality \beth_\alpha for all \alpha<\omega_1^{CK}, then it has models of all infinite cardinalities.
Theorem 2: (Using Kleene’s O:) For every ordinal notation a\in O, there exists a computable structure A, such that A characterizes \beth_{|a|}, where |a| is the ordinal defined by a.
Combining the above two theorems we obtain that the Hanf number for Scott sentences of computable structures is equal to \beth_{\omega_1^{CK}}. This answers a question of Sy D. Friedman.

Andrés Caicedo: Topological partition calculus of countable ordinals

Thursday, April 14, 2016; 4:00-5:30 PM, in East Hall 2866.

This is joint work with Jacob Hilton. We considered the topological version of the partition calculus in the setting of countable ordinals: Given ordinals $\alpha,\beta_0,\beta_1$, we say that $\alpha\to_{top}(\beta_0,\beta_1)^2$ iff for any 2-coloring of the edges of the complete graph on $\alpha$ vertices, for some color $i$, there is a complete monochromatic graph in color $i$ whose set of vertices is homeomorphic to $\beta_i$. If we insist that $\alpha,\beta_0$ are countable and that $\beta_0>\omega$, then $\beta_1$ must be finite (even if we only require order-types rather than homeomorphisms). On the other hand, we have proved that for any countable $\beta_0$ and finite $\beta_1$, we can find a countable $\alpha$ such that $\alpha\to_{top}(\beta_0,\beta_1)^2$. This is a topological version of the Erdös-Milner theorem. Our arguments provide explicit bounds. I will discuss some of these results.