Dates: 01 September 2021 – 29 February 2024

Details on EPSRC grant EP/V009001/1

- Principal Investigator: Philip Welch
- Researcher: Philipp Schlicht

While many questions about the structure of abstract sets are not decided by the axioms of set theory, descriptive set theory provides a rich structure theory of sufficiently simple subsets of Polish spaces, i.e. countably based complete metric spaces. It began with the work of analysts like Borel and Lebesgue and later led to important developments in set theory and applications in measure theory, functional analysis and dynamics.

A number of results on graph colorings have emerged more recently from work of Kechris, Solecki, Todorcevic [KST99] and Miller [Mil12, CMS]. These state that a sufficiently simple graph or hypergraph can either be coloured in few colors or contains a canonical counterexample from a finite list. The central importance of these theorems stems from the fact that they imply many previous results.

Generalised descriptive set theory, a relatively recent field, develops analogues to results and techniques from descriptive set theory for generalised Baire spaces of functions on an uncountable cardinal equipped with a natural topology. It is in part motivated by the work of model theorists studying the classification theory of uncountable models [FHK14] and recently became very active and because of its deep connections with other areas such as infinite combinatorics, games and large cardinals.

In this project, we establish results about hypergraphs on generalised Baire spaces using infinite combinatorics, large cardinals and games. As applications, we aim to lift some theorems from descriptive set theory and analysis to the setting of generalised Baire spaces.

Project publications:

- Peter Holy, Philipp Schlicht, Christopher Turner, Philip D. Welch. Asymmetric cut and choose games. 31 pages. To appear in
*Bulletin of Symbolic Logic,*2023 -
Philipp Schlicht, Christopher Turner. Forcing axioms via ground model interpretations. 45 pages. To appear in

*Annals of Pure and Applied Logic,*2023 - Sandra Müller, Philipp Schlicht. Uniformization and Internal Absoluteness. 13 pages. To appear in
*Proceedings of the American Mathematical Society,*2022 -
Peter Holy, Marlene Koelbing, Philipp Schlicht, Wolfgang Wohofsky. Ideal topologies in higher descriptive set theory.

*Annals of Pure and Applied Logic 173, 4, 103061*, 36 pages, 2022 -
Merlin Carl, Philipp Schlicht. Canonical Truth. Axiomathes 32, 785–803, 2022

**References**

[KST99] Kechris, Alexander S., Slawomir Solecki, and Stevo Todorcevic. Borel chromatic numbers.

*Advances in Mathematics*141, 1, 1-44, 1999[Mil12] Benjamin D Miller. The graph-theoretic approach to descriptive set theory.

*Bulletin of Symbolic Logic*, 554–575, 2012[CMS] Raphaël Carroy, Benjamin D. Miller, and Dániel T. Soukup. The open dihypergraph dichotomy and the second level of the Borel hierarchy.

*arXiv preprint arXiv:1803.03205,*2018[FHK14] Sy-David Friedman, Tapani Hyttinen, and Vadim Kulikov. Generalized descriptive set theory and classification theory.

*Memoirs of the American Mathematical Society*, 2014