One of the cornerstones of the forcing technique is the surprisingly elementary R-S lemma, which provides us with the existence of generic filters in ZFC. In this talk, we will discuss some original proofs of known results in an attempt to simplify some of the classical arguments, as well as some mention of a work in progress.
The R-S lemma can be used to construct order isomorphisms as a more illustrative alternative to the back-and-forth paradigm, yielding characterizations of the rationals, the reals, countable atomless boolean algebras and even some forcing notions, naturally seen as ordered structures.
Extending this idea, we will also show how to build graph isomorphisms that (almost always) characterize countable infinite random graphs. Moreover, if there is time, we will increase the sophistication of the forcing notions we use so that the R-S lemma produces large homogeneous sets, proving the celebrated theorems of Ramsey and Hindman.
The cherry on top of our perhaps overly ambitious cake will be a sketch of how the R-S lemma can be seen as a type of compactness/selection principle, resulting in finite variants of our favourite theorems in Ramsey theory.
A structure M is said to be pseudofinite if every first-order sentence that is true in M has a finite model, or equivalently, if M is elementarily equivalent to an ultraproduct of finite structures. For this kind of structures, the fundamental theorem of ultraproducts ( Los’ Theorem) provides a powerful connection between finite and infinite sets, which can sometimes be used to prove qualitative properties of large finite structures using combinatorial methods applied to non-standard cardinalities of definable sets.
The concept of measurable structures was defined by Macpherson and Steinhorn in [3] as a method to study infinite structures with strong conditions of finiteness and definability for the sizes of definable sets. The most notable examples are the ultraproducts of asymptotic classes of finite structures (e.g., the class of finite fields or the class of finite cyclic groups). Measurable structures are supersimple of finite SU-rank, but recent generalizations of this concept are more flexible and allow the presence of structures whose SU-rank is possibly infinite.
The everywhere infinite forest is the theory of an acyclic graph G such that every vertex has infinite degree. It is a well-known example of an omega-stable theory of infinite rank. In this talk we will take this structure as a motivating example to introduce all the concepts mentioned above, showing that it is pseudofinite and giving a precise description of the sizes of their definable sets. In particular, these results provide a description of forking and U-rank for the infinite everywhere forest in terms of certain pseudofinite dimensions, and also show that it is a generalized measurable structure that can be presented as the ultraproduct of a multidimensional exact class of finite graphs.
Given a tree T, the sibling number of T is the number of isomorphism classes of trees mutually embeddable and non-isomorphic with T. The Tree Alternative Conjecture states that the sibling number of any tree is either 1 or infinite. The conjecture has been verified for rayless, rooted and stable trees. Tyomkyn made some progress towards the conjecture for locally finite trees and conjectured that if a locally finite tree has a non-surjective self-embedding, then its sibling number is infinite.
I will present two approaches towards both conjectures for locally finite trees. The direction of a self-embedding is defined based on a canonical concept namely end. Hamann proved that a tree has either 0, 1, 2 or infinitely many directions. The last possible size provides interesting and challenging properties for which the conjecture is still open.
Leaf decomposition is a useful technique in which rooted components with leaves play a key role in determining the sibling number and is useful even if the tree is not locally finite.
In 1974 N. Hindman proved that for any finite colouring of the natural numbers, there is an infinite subset X such that all finite sums of elements of X have the same colour. This has a natural reinterpretation in terms of the set FIN of all finite subsets of N: For every finite colouring of FIN, there is a block sequence such that all finite unions of members of such a sequence have the same colour. In this talk, we will give a quick proof of Hindman's theorem using ultrafilter techniques before discussing various generalizations, including some recent developments. Ideas behind the proofs of each such generalization of Hindman's theorem will be discussed.
Some results in the theory of star selection principles are about the size of closed discrete subsets of spaces with one of these star selection properties. On one hand, Sakai showed that if X is strongly star-Menger and Y is a closed discrete subset of X, then the size of Y is less than the continuum. On the other hand, da Silva showed that if X is a separable selectively (a)-space and Y is a closed discrete subset of X, then the size of Y is less than the continuum. Since absolutely strongly star-Menger implies any of the properties in those results, we get that closed discrete subsets of (separable) absolutely strongly star-Menger spaces are of size less than the continuum.
In this talk, we will show that in fact, closed discrete subsets of separable absolutely strongly star-Menger spaces are of size less than the dominating number and the analogous case for the Hurewicz-type properties. Also, we will mention some results that involve closed discrete subsets of small spaces satisfying some other related star selection principles.
This is joint work with Sergio Garcia Balan.
An important result in the theory of star selection principles, shown by Bonanzinga and Matveev in 2009, states that a Psi-space is strongly star-Menger if and only if the size of the associated almost disjoint family is smaller than the dominating number. Five years later, Sakai generalized one side of this result showing that every strongly star-Lindelof space of size smaller than the dominating number it is strongly star-Menger (observe that all Psi-spaces are strongly star-Lindelof).
In this talk we will present some variations of this result and we will mention as well the analogues when you replace "Menger" by "`Hurewicz" or "Rothberger".
We will provide definitions, diagrams and pictures.
This is joint work with Javier Casas de la Rosa.
February 09: Daniel Calderón (University of Toronto)
Borel's conjecture states that every strongly null set of real numbers should be countable. A classical result of Galvin, Mycielski, and Solovay, states that Borel's strong nullity can be rephrased as an algebraic property for subsets of the line: a subset X of the reals is strongly null if, and only if, for every meager subset M of the reals, X+M is not the whole line. A subset of the reals is called meager-additive if X+M is meager for every meager subset M of R. Clearly, all meager-additive sets are strongly null. The following problem is due to Bartoszynski and Judah:
Suppose every strongly null set of real numbers is meager-additive. Does Borel's conjecture follow?
In recent work, I solved this question for the negative by introducing a family of forcing notions P(t) that are taste-wise very similar to Silver's forcing and are such that in the generic extension obtained after a countable support iteration of P(t)'s, every strongly null subset of the line is meager-additive, and there are uncountable meager-additive sets.
In this talk we are going to try to understand Fraisse theory from an alternate viewpoint. Rather than try to prove the amalgamation property, we are going to check for the weaker requirement that there is a winning strategy to a variant of the Banach Mazur game. We will explain why this is indeed weaker and how this game categorizes an alternative type of property called weak Fraisse. While the property may be weaker, the game theoretic interpretation enables us to better understand (and potentially classify) the (weak) Fraisse limit. Given the natural question to ask after proving a class is Fraisse is "What the heck do I know about the limit that's meaningful outside of ultrahomogeneity?" this talk will be beneficial to anyone interested in studying Fraisse classes and their limits. However, the talk will be incredibly rudimentary (no deep new soon to be published results here.... Well, maybe a peek at one 😉.). Anyone with familiarity of basic finite structures and a passing knowledge of set theory should be able to understand the talk.