# Partitions of well-founded trees and two connections with model theory. Paris (v) – 6/10

For the Paris Logic Seminar, I gave the lecture

Abstract: in 2003, Komjath and Shelah proved a partition theorem on scattered order types; these in turn could be understood as partition relations for classes of well-founded trees. Recently, two different kinds of applications of the same partition relation have been used in infinitary logic and in model theory: one by Väänänen and Velickovic on games related to Shelah’s logic $L^1_\kappa$, another by Shelah and myself on the “canonical tree” of an AEC (a generalization of the Scott sentence for an abstract elementary class). I will describe the Komjath-Shelah result in the first part and then narrow in the applications (with more details on the second one, from some recent joint work with Shelah).

# One Puzzling Logic, Two Approximations… and a Bonus. Helsinki (v), 5/20

For the Helsinki Logic Seminar, I gave (virtually) the lecture

One Puzzling Logic, Two Approximations… and a Bonus.

Abstract: The puzzling logic (called $L^1_\kappa$ for $\kappa$ a singular strong limit cardinal) I will speak about was introduced by Saharon Shelah in 2012. The logic $L^1_\kappa$ has many properties that make it very well adapted to model theory, despite being stronger than$L_{\kappa,\omega}$. However, it also lacks a good syntactic definition.

With Väänänen, we introduced the first approximation (called $L^{1,c}_\kappa$,) as a variant of $L^1\kappa$ with a transparent syntax and many of the strong properties of Shelah’s logic.

The second approximation (called Chain Logic), while not new (it is due to Karp), has been revisited recently by Dzamonja and Väänänen) also in relation to Shelah’s $L^1_\kappa$ and the Interpolation property.

I will provide a description of these three logics, with emphasis on their relevance to model theory.
As a bonus, I will make a connection between these logics and axiomatizing correctly an arbitrary AEC. This last part is joint work with Shelah.