Compactness Principles and Combinatorics

In this project focus on the interplay between two different areas of set theory, which is expected to reveal properties of the set theoretic, and so more broadly mathematical universe which each of the two areas separately seem to be insufficient in its techniques and methods to address. A natural intersection of the two areas is the generalized Baire space, which can be described as a generalization of the real line into the higher infinite.

This project studies infinitary combinatorial properties of the so called higher Baire spaces. The higher Baire spaces are a natural analogue of the classical Baire spaces at higher cardinals. The project focuses on central problems of the area and offers a number of groundbreaking results, including: the study of higher Baire spaces analogues of the classical independent families; the development of a new forcing iteration technique, which can be described as a system of parallel non-linear iterations; a close analysis of the relation between the higher Baire spaces analogues of the tower and pseudo-intersection numbers, building an analogue with the celebrated result of Malliaris-Shelah that the two invariants in the countable setting coincide; as well as giving a global evaluation of the spectrum of kappa maximal almost disjoint families. The obtained results and developed techniques mark a significant advancement in this actively growing area of set theory. The developed theory forms a firm foundation towards approaching important remaining open problems.