As for this theorem, several versions can be considered, imposing constraints on cardinalities of languages, cardinalities of structures, and cardinalities of underlying sets of ultrafilters. In this talk, we will examine the relationship between these versions and cardinal invariants.

For a Hausdorff space $X$ let $\mathcal{P}$ be a family of subsets of $X$. Player I and Player II play on $X$ a game $G_{\mathcal P}$ as follows. Player I starts and on each stage $n\in\omega$ chooses a subset $X_n \in \mathcal{P}$ such that $X_0 = \emptyset$ and every $X_n$ is disjoint with all previously chosen sets $\langle X_i \mid i \leq n-1 \rangle$. Player II responds at stage $n\in\omega$ by choosing an open set $U_n \subseteq X$ such that $X_n \subseteq U_n$ for every $n\in\omega$. So, playing $G$, the players produce a sequence of pairs $\langle (X_n, U_n) \mid n \in \omega \rangle$.

Player II wins if after $\omega$ moves the family of open sets $\langle U_n \mid n \in \omega \rangle$ is point-finite; otherwise, Player I wins.

Definition 2. Denote by $G(\Delta)$ the game $G_{\mathcal P}$ where $\mathcal{P}$ is the family of all subsets of $X$; denote by $G(\Delta_0)$ the game $G_{\mathcal P}$ where $\mathcal{P}$ is the family $\{\emptyset\}\cup\{\{x\}\mid x \in X\}$; denote by $G(\Delta_1)$ the game $G_{\mathcal P}$ where $\mathcal{P}$ is the family consisting of all countable subsets of $X$; denote by $G(\Delta_2)$ the game $G_{\mathcal P}$ where $\mathcal{P}$ is the family consisting of all compact subsets of $X$.

Problem 1. Find characterizations of the classes of topological spaces $X$ for which the games $G(\Delta)$, $G(\Delta_2)$, $G(\Delta_1)$, and $G(\Delta_0)$ are determined.

Problem 2. Is the game $G(\Delta)$ determined, assuming that $X$ is a $\Delta$-set or a $Q$-set of reals?

I shall discuss known results in connection with this family of axioms, and open problems in the context.

Assuming CH, he moreover obtained what is now known as a ""Shelah group"" of size $\aleph_1$, i.e., an uncountable group such that for some integer $N$, the collection of all $N$-sized words over the alphabet of any given uncountable subset of the group resurrects the whole group. In this talk, we shall present a ZFC construction of a Shelah group at the level of any successor of a regular cardinal. We shall also address the problem of constructing Shelah groups at successors of singulars and at inaccessibles. This is joint work with A. Rinot.

These will organized as internal and external developments. Internal developments will the contrast the role of a local and global approach and, in particular, the importance of frames and tameness in the two approaches. These routes merge in the Shelah-Vasey categoricity paper. External progress will consider the connections with accessible categories and the eventual categorical definition of forking, the role of large cardinal axioms and the weak generalized continuum hypothesis, and applications to the theory of modules and interactions with algebraic geometry and number theory.

This is a joint work with Balko, Chodounský, Dobrinen, Hubička, Konečný, Nešetřil, Todorcevic, Unger and Zucker

If there is time, I will argue that which universally Baire notion is dual to the universally measurable sets.

In this talk is I give several results about discrete density, as well as other similar cardinal functions.

Here is a short list of some of these results:

If $X$ is $T_2$ then $Dd(X) \le 2^{w(X)}$ and if $X$ is $T_3$ then $Dd(X) \le 2^{nw(X)}$. It is consistent to have $X \subset 2^{\omega_1}$ s.t. $\vert X\vert = Dd(X) = \omega_2$. $d(2^\kappa) = Dd(2^\kappa) = \log \kappa$ for every cardinal $\kappa \ge \omega$. $Dd(\beta Q) = 2^\omega$.

This is joint work with Alan Dow and Jan van Mill.