# Tutorial Track

## C. Lambie-Hanson: Nontrivial coherent families of functions

In this tutorial, we will survey the history of and recent developments in the set theoretic study of nontrivial coherent families of functions, focusing in particular on families indexed by the space ${^\omega}\omega$. Such families originally arose out of homological considerations (they can be seen as witnesses to the nontriviality of certain derived inverse limits, or as witnesses to the nonadditivity of strong homology), but they can also be thought of as purely set theoretic objects of interest in their own right.

In the first half of the tutorial, we will cover work from the late 1980s and early 1990s connecting the existence of 1-dimensional nontrivial coherent families with topics such as cardinal characteristics of the continuum and the Open Coloring Axiom. In the second half, we will cover some recent results about higher-dimensional nontrivial coherent families. The focus of the tutorial will be on the set theoretic aspects of the topic, but we will also touch on its origins in and continued applications to homological algebra.

## J. Hubička: Big Ramsey degrees using parameter spaces

We show that the universal homogeneous partial order has finite big Ramsey degrees and discuss several corollaries. Our proof uses parameter spaces and the Carlson-Simpson theorem rather than (a strengthening of) the Halpern-Läuchli theorem and the Milliken tree theorem, which are the primary tools used to give bounds on big Ramsey degrees elsewhere (originating from work of Laver and Milliken). This new technique has many additional applications. To demonstrate this, we show that the homogeneous universal triangle-free graph has finite big Ramsey degrees, thus giving a short proof of a recent result of Dobrinen.

## M. Sabok: Hyperfinite graphs

In this tutorial, we will start with an introduction to hyperfinite graphs defined on probability spaces and then discuss several combinatorial problems arising in the setting of measurable combinatorics and measured group theory.

In particular, we will discuss the existence of measurable perfect matchings in hyperfinite graphs and we will see that bipartite graphs that are regular and one-ended always admit such perfect matchings. We will give some applications of the latter results to equidecomposition problems, such as the measurable circle squaring, and the existence of factors of iid perfect matchings in finitely generated groups.

## O. Zindulka: Strong Measure Zero: Geometry and Combinatorics

The history of Strong measure zero spans more than a century. We will give a very brief account of classical results and focus on more recent research.

We will see how strong measure zero is related to Hausdorff measures and dimension, Borel submeasures and measures, Ramsey Theory, Game Theory and combinatorics of open covers. Special attention will be paid to the celebrated Galvin-Mycielski-Solovay Theorem and its pitfalls in other Polish groups.

Galvin-Mycielski-Solovay Theorem inspired the notion of meager-additive sets. We will look into parallels of meager-additive sets and strong measure zero - characterization by fractal measures, Game Theory, Ramsey Theory, covers etc. We will also show that a set $X$ in the Cantor set is meager-additive if and only if for each closed null set $E$ the algebraic sum $X+E$ is closed null.

There are other similar notions (null-additive sets and two more). I will walk you through a theory based on the selection principle $\mathsf S_1$ unifying all of the five notions.