**Lecture**

Ehrenfeucht-FraÏssé Games in the Uncountable Environment: Where is the Scott Watershed?

Rahman Mohammadpour, Institut für Diskrete Mathematik und Geometrie- TU Wien

7 DEC 2023

11:00 - 12:30

The Ehrenfeucht-Fraïssé games and their variants are a natural game-theoretic tool for investigating the similarity and dissimilarity between two mathematical structures. Although the study of these games is quite different in finite and infinite model theory, one might ask: if not all natural numbers are equal, why should all infinities be equal? In fact, it is within the framework of countably infinite model theory, not infinite model theory, that these games are often successful!

Even at the level of the first uncountable cardinal, the picture is very mixed, but this is not surprising: The Sikorski extension theorem vs. non-trivial forcing axioms, countability and uncountability in Shelah's classification theory, descriptive set theory vs. generalised descriptive set theory.

In this series of lectures (for some n> 1) I will talk about an area of research that started in the early 1990s and is an intersection of several independent topics.

The lectures will start with a brief introduction to infinite games, then continue with an introduction to the classical Ehrenfeucht-Fraïssé games, and then look at these games for uncountable structures. We will see how they lead to large Aronszajn trees and the notion of weak embeddability between them. The notion of weak embeddability between ordered structures goes back to the early post-Cantor era (e.g. in a 1915 paper by Hartogs), possibly in a different terminology. The notion has attracted attention in various mathematical contexts after being explored and popularised by Todorcevic, Mekler and Väänänen. Part of the talk will be devoted to this topic. I will discuss the concept of weak embeddability between ordered structures, emphasising its historical significance and mathematical importance. I will mention recent progress on the famous problem of a universal tree in the class of (wide) Aronszajn trees. If time permits, I will also present a proof sketch showing that the universality number of (wide) Aronszajn trees is never the first uncountable cardinal number.

**Venue**: Niavaran, Lecture Hall 2