# Tutorial Track

## C. Laflamme: Topics in Combinatorial Set Theory

#### Symmetries of some Homogeneous Structures

slidesAutomorphisms may not describe all the possible "symmetries" of the structure. As an example and exercise can you describe all functions on the rationals which map all copies to copies? Homogeneous structures are "highly" symmetric in the sense that any partial finite automorphisms extends to an automorphisms, and we investigate other symmetries for the Rado graph and other homogeneous structures.

#### How many siblings do you have?

slidesTwo structures A and B are siblings if each embeds in the other. As an example and exercise can you count / describe all siblings of the rationals (up to isomorphism)? We will discuss the case of all countable linear orders, and conclude with a discussion of the more general case of trees and graphs.

#### Is converging bounded sequences really harder than converging binary sequences?

slidesThis is a purely set theoretical problem originating from the work of Vojtas and others in the late 80's; in his notation the question is whether $r=r_\sigma$. As an exercise can you show that these cardinals are indeed equal? I will review some of the history of this problem, exhibiting the frustratingly low level of understanding we still have on this basic combinatorial problem, but will try to conclude with an optimistic outlook.

## D. Milovich: Topological applications of long $\omega_1$-approximation sequences

slides I, slides II, slides IIIEvery ordinal has a uniformly definable cardinal normal form analogous to its Cantor normal form. As a consequence, given any transfinite sequence of countable elementary substructures of (a large fragment of) the set-theoretic universe, the union of the range of the sequence is a finite union of (possibly uncountable) elementary substructures, provided the sequence satisfies the very weak requirement that every structure in the sequence contains as an element (but not necessarily as a subset) the subsequence of all preceding structures. Such sequences can be loosely thought of as extremely coarse morasses available in ZFC.

The original, implicit application of cardinal normal form is a 1964 ZFC proof by R. O. Davies that the Euclidean plane is a countable union of rotated graphs of functions. (A much earlier construction by Sierpinski required the Continuum Hypothesis.) A 2008 topological application is that every compact group has an open-in-finite base, that is, a topological base such that every nonempty open subset of the group has at most finitely many supersets in the base.

The tutorial will review basic facts about elementary substructures, introduce the cardinal normal form and long $\omega_1$-approximation sequences, prove the fundamental finite union property of long $\omega_1$-approximation sequences, prove the two mentioned applications, and conclude with a brief discussion of more recent and potential future applications.

## J. T. Moore: Iterated forcing and the Continuum Hypothesis

slides I, slides II, slides IIIOne of the most fundamental questions one can ask about an iterated forcing construction is whether it introduces new real numbers. Clearly it is necessary that the iterands themselves do not introduce new reals, but it is well known that this is not sufficient. This tutorial will develop the theory of iterated proper forcing in the presence of CH through a specific construction: we will show how to construct a model of ZFC in which $\omega_1$ and $-\omega_1$ are the only minimal uncountable order types.

## A. Roslanowski: Properness for iterations with uncountable supports

slides I, slides II, slides IIIWhile there are still open problems in the theory of forcing iterated with finite and/or countable support and we still need to expand our preservation theorems, there is a sense that we understand these iterations pretty well. We use them frequently, even if we tend to skip iterated forcing in favour of the respective forcing axioms (MA or PFA).

It is natural to look at iterations with uncountable
supports and ask for parallel theorems. However, it has been known
since the beginning of 1980s that there is no straightforward theory
of iterations with uncountable supports and cardinals are frequently
collapsed in such iterations. Still, there are in literature also
positive results concerning not collapsing cardinals in iterations
with uncountable supports. For instance, in 1980, Kanamori considered
$\lambda$-support iterations of $\lambda$-Sacks forcing notion and
he
proved
that under some circumstances these iterations preserve
$\lambda^+$. In 2003, Eisworth
discussed
iterated forcing for
successors of regular cardinals introducing a variant of Shelah's
*properness over semi-dimonds*. Fusion properties of iterations of
tree-like forcing notions were used in by
Friedman and Zdomskyy
in 2010, and then by
Friedman, Honzik and Zdomskyy
in 2013.

In a series of articles (1,2,3,4,5,6,7) by Shelah and myself we introduced and studied several strong versions of $\lambda$-properness appropriate for iterations with supports of size $\leq\lambda$ of natural forcing notions adding a new member of ${}^\lambda\lambda$ without adding new elements of ${}^{ {<}\lambda} \lambda$. The tutorial will present the motivation, highlights and dependencies among those properties. We will also look at some of the relevant iteration theorems.