The partition relation A→(P)^μ_λ, despite its brevity, is remarkably expressive. This fundamental combinatorial principle asserts that every λ-coloring of μ-sized subsets of A is constant on a subset in the class P. By adjusting the parameters A, μ, λ, and P, one can express a wide variety of large cardinal principles, including weak compactness, Ramsey-ness, and even supercompactness. In this talk, we focus on the case where A is an uncountable cardinal κ and P is the class of stationary subsets of κ. By work of Baumgartner 1977, it turns out that this corresponds to certain ineffability properties of κ. We also describe how this fits into a different hierarchy of ineffability properties described in current joint work with Matthew Foreman and Menachem Magidor.

This is the second in a series of lectures going through notes entitled "Naive Descriptive Set Theory" that are available on ArXiV.  

In the last 20 years the field has had many applications to areas in Analysis and Dynamical Systems The lectures are intended to be an opportunity to learn the subject matter, and will be interspersed with research lectures during the quarter.

No background beyond basic elements of the 210 sequence are required. 
 

The Ultrapower Axiom (UA) roughly states that any pair of ultrapowers can be compared by internal ultrapowers. The Axiom was extensively studied by Gabriel Goldberg, leading to a series of striking results.

Goldberg asked whether UA is consistent with a measurable cardinal that violates GCH. The main challenge is that UA is not easily preserved under forcing constructions, especially ones that achieve violation of GCH on a measurable from large cardinal assumptions. For example, such forcings might create normal measures which are incomparable in the Mitchell order – a property that negates UA.
In this talk, we sketch the proof that the failure of GCH on the least measurable cardinal can indeed be forced while preserving UA, starting from the minimal canonical inner model carrying a (\kappa, \kappa^{++})-extender. We will present the forcing construction and sketch the main proof ideas. This is a joint work with Omer Ben-Neria.

 It is well-known that a finite set is universal, that is, each Lebesgue measurable set with positive measure contains an affine copy of a finite set. The Erdős similarity conjecture, which remains open, states that there is no infinite universal set. In 2022, Gallagher, Lai, and Weber considered a topological version of this conjecture, defining a set to be topologically universal if each dense G-delta set contains an affine copy of the set. They conjectured that there are no such uncountable sets. In this thesis, we give a full classification of topologically universal sets as a special subfamily of measure zero sets. As a corollary, we prove that the topological Erdős similarity conjecture is independent of ZFC. We generalize this result to arbitrary locally compact Polish groups, and use the measure-category duality to pose and investigate the full-measure Erdős similarity conjecture.

The axiom AD^+, a structural strengthening of the Axiom of Determinacy (AD), was introduced by Hugh Woodin in the 1980's. AD^+ resolves many basic structural questions unsettled by AD. However, there are still many basic questions not answered by AD^+. One such class of questions concerns comparing cardinalities of sets under AD^+: given any two sets X and Y, how can we compare |X| and |Y|? One concrete instance of this is the following conjecture. 

Conjecture (the ABCD conjecture): suppose \alpha,\beta,\gamma,\delta are infinite cardinals such that \beta \leq \alpha and \delta\leq \gamma. Then |\alpha^\beta| \leq |\gamma^\delta| if and only if \alpha\leq \gamma and \beta \leq \delta.

The ABCD Conjecture is false under ZFC. It is open whether AD^+ implies the conjecture holds, but many instances of the conjecture have been established (by work of Woodin, Chan-Jackson-Trang etc). We introduce a structural strengthening of the axiom AD^+, called AD^{++}. AD^{++} implies the ABCD Conjecture and appears to have other interesting consequences not known to follow from AD^+. We do not know if AD^+ implies AD^{++} but some special cases have been proved. We will define these notions and discuss some of the partial results mentioned above. This is ongoing joint work with W. Chan and S. Jackson.