By the Cantor-Bendixson theorem, subtrees of the binary tree on $\omega$ satisfy a dichotomy - either the tree has countably many branches or there is a perfect subtree (and in particular, the tree has continuum manybranches, regardless of the size of the continuum). We generalize this to arbitrary regular cardinals $\kappa$ and ask whether every $\kappa$-tree with more than $\kappa$ branches has a perfect subset. From large cardinals, this statement isconsistent at a weakly compact cardinal $\kappa$. We show using stacking mice that the existence of a non-domestic mouse (which yields a model with a proper class of Woodin cardinals and strong cardinals) is a lower bound. Moreover, we study variants of this statement involving sealed trees, i.e. trees with the property that their set of branches cannot be changed by certain forcings, and obtain lower bounds for these as well. This is joint work with Yair Hayut.[-]

Two-player turn-based zero-sum games on (finite or infinite) graphs are a central framework in theoretical computer science — notably as a tool for controller synthesis, but also due to their connection with logic and automata theory. A crucial challenge in the field is to understand how complex strategies need to be to play optimally, given a type of game and a winning objective. I will give a tour of recent advances aiming to characterize games where finite-memory strategies suffice (i.e., using a limited amount of information about the past). We mostly focus on so-called chromatic memory, which is limited to using colors — the basic building blocks of objectives — seen along a play to update itself. Chromatic memory has the advantage of being usable in different game graphs, and the corresponding class of strategies turns out to be of great interest to both the practical and the theoretical sides.[-]

An atomic class $K$ is the class of atomic first order models of a countable first order theory (assuming there are such models). Under the weak $\mathrm{GCH}$ it had been proved that if such class is categorical in every $\aleph_n$ then it is categorical in every cardinal and is so called excellent. There are results when we assume categoricity for $\aleph_1, \ldots, \aleph_n$. The lecture is on a ZFC result in this direction for $n=1$. More specifically, if $K$ is categorical in $\aleph_1$ and has a model of cardinality $>2^{\aleph_0}$, then it is $\aleph_0$-stable, which implies having stable amalgamation, and is the first case of excellence.
This a work in preparation by J.T. Baldwin, M.C. Laskowski and S. Shelah.[-]

N. Hindman, I. Leader and D. Strauss proved that if $2^{\aleph_0}<\aleph_\omega$ then there is a finite colouring of $\mathbb{R}$ so that no infinite sumset $X+X$ is monochromatic. Now, we prove a consistency result in the other direction: we show that consistently relative to a measurable cardinal for any $c:\mathbb{R}\to r$ with $r$ finite there is an infinite $X\subseteq \mathbb{R}$ so that $c\upharpoonright X+X$ is constant. The goal of this presentation is to discuss the motivation, ideas and difficulties involving this result, as well as the open problems around the topic. Joint work with P. Komjáth, I. Leader, P. Russell, S. Shelah and Z. Vidnyánszky.[-]

Recent work has clarified how various natural second-order set-theoretic principles, such as those concerned with class forcing or with proper class games, fit into a new robust hierarchy of second-order set theories between Gödel-Bernays GBC set theory and Kelley-Morse KM set theory and beyond. For example, the principle of clopen determinacy for proper class games is exactly equivalent to the principle of elementary transfinite recursion ETR, strictly between GBC and GBC+$\Pi^1_1$-comprehension; open determinacy for class games, in contrast, is strictly stronger; meanwhile, the class forcing theorem, asserting that every class forcing notion admits corresponding forcing relations, is strictly weaker, and is exactly equivalent to the fragment $\text{ETR}_{\text{Ord}}$ and to numerous other natural principles. What is emerging is a higher set-theoretic analogue of the familiar reverse mathematics of second-order number theory.[-]

It is well-known that the statement "all $\aleph_1$-Aronszajn trees are special'' is consistent with ZFC (Baumgartner, Malitz, and Reinhardt), and even with ZFC+GCH (Jensen). In contrast, Ben-David and Shelah proved that, assuming GCH, for every singular cardinal $\lambda$: if there exists a $\lambda^+$-Aronszajn tree, then there exists a non-special one. Furthermore:
Theorem (Ben-David and Shelah, 1986) Assume GCH and that $\lambda$ is singular cardinal. If there exists a special $\lambda^+$-Aronszajn tree, then there exists a $\lambda$-distributive $\lambda^+$-Aronszajn tree.
This suggests that following stronger statement:
Conjecture. Assume GCH and that $\lambda$ is singular cardinal.
If there exists a $\lambda^+$-Aronszajn tree,
then there exists a $\lambda$-distributive $\lambda^+$-Aronszajn tree.

The assumption that there exists a $\lambda^+$-Aronszajn tree is a very mild square-like hypothesis (that is, $\square(\lambda^+,\lambda)$).
In order to bloom a $\lambda$-distributive tree from it, there is a need for a toolbox, each tool taking an abstract square-like sequence and producing a sequence which is slightly better than the original one.
For this, we introduce the monoid of postprocessing functions and study how it acts on the class of abstract square sequences.
We establish that, assuming GCH, the monoid contains some very powerful functions. We also prove that the monoid is closed under various mixing operations.
This allows us to prove a theorem which is just one step away from verifying the conjecture:

Theorem 1. Assume GCH and that $\lambda$ is a singular cardinal.
If $\square(\lambda^+,<\lambda)$ holds, then there exists a $\lambda$-distributive $\lambda^+$-Aronszajn tree.
Another proof, involving a 5-steps chain of applications of postprocessing functions, is of the following theorem.

Theorem 2. Assume GCH. If $\lambda$ is a singular cardinal and $\square(\lambda^+)$ holds, then there exists a $\lambda^+$-Souslin tree which is coherent mod finite.

This is joint work with Ari Brodsky. See: http://assafrinot.com/paper/29[-]

The Galvin-Prikry theorem states that Borel partitions of the Baire space are Ramsey. Thus, given any Borel subset $\chi$ of the Baire space and an infinite set $N$, there is an infinite subset $M$ of $N$ such that $\left [M \right ]^{\omega }$ is either contained in $\chi$ or disjoint from $\chi$ . In their 2005 paper, Kechris, Pestov and Todorcevic point out the dearth of similar results for homogeneous relational structures. We have attained such a result for Borel colorings of copies of the Rado graph. We build a topological space of copies of the Rado graph, forming a subspace of the Baire space. Using techniques developed for our work on the big Ramsey degrees of the Henson graphs, we prove that Borel partitions of this space of Rado graphs are Ramsey.[-]