# STUK 3

### 3 October 2019, School of Mathematics, University of Leeds

The third meeting of the LMS Set Theory in the UK network was held at the University of Leeds on Thursday the 3rd of October, 2019. The speakers we Asaf Karagila, David Aspero and Heike Mildenberger. All were welcome.

### Time and Venue

The meeting started with tea and coffee at 10 am, with talks starting at 10:30. Talks were be held in the "MALL" on floor 8 of the School of Mathematics. Tea and coffee were be served on floor 9.

For information on how to reach the School of Mathematics, please read the University instructions or see our location on Google Maps.

### Schedule and slides

 10:00 Tea and coffee 10:30 Asaf Karagila: Preserving bits of choice 11:40 David Aspero: Generalizations of Martin’s Axiom, square, and a forcing axiom failure 12:40 Lunch 2:15 Heike Mildenberger: Local Ramsey Spaces in Matet Forcing Extensions 3:15 Tea and coffee 4:00 Questions and discussion session: Philipp and Andrew's questions 5:00 Pub: The Fenton 6:00 Dinner: Red Chilli

### Abstracts

#### Preserving bits of choice

Asaf Karagila, University of East Anglia

Violating the axiom of choice is easy, but preserving just some of it is hard. We will introduce some criteria that can help with that, and discuss some related problems.

#### Generalizations of Martin’s Axiom, square, and a forcing axiom failure

David Aspero, University of East Anglia

I will present consistent forcing axioms extending Martin’s Axiom at lambda for arbitrary lambda, will prove that one of them implies $\Box_{\omega_1}$, and will use this result to show that a natural extension of Martin’s Axiom is inconsistent.

#### Local Ramsey Spaces in Matet Forcing Extensions

Heike Mildenberger, University of Freiburg

We introduce Gowers-Matet forcing with a finite sequence of pairwise non-isomorphic Ramsey ultrafilters over the natural numbers, and with this forcing we settle the long-standing problem about the spectrum of numbers of near-coherence classes. For evaluating the new forcing, we prove a strengthening of Gowers' theorem on colourings of the k-valued block sequences.

### Travel expenses

The meeting is generously supported by the London Mathematical Society and the School of Mathematics of the University of Leeds. As such, we have some funds available to reimburse participants' travel expenses, including hotel costs the night before and/or after - contact Andrew Brooke-Taylor at a.d.brooke-taylor@leeds.ac.uk for details.