Borel Combinatorics Seminar


Title: An invitation to Borel Polymorphisms

Time: Thursday, 29.02.2024, 16:15.
Place: ELTE, D3-306.
Abstract: I will discuss in some detail why is it hard to solve Borel systems of linear equations over finite fields and the relationship of these results to Borel polymorphisms.

Matt Bowen (Oxford)

Title: Monochromatic products and sums

Time: Thursday, 22.02.2024, 16:00.
Place: ELTE, D3-306.
Abstract: We show that any 2-coloring of the naturals and any finite coloring of the rationals contains many monochromatic sets of the form {x, y, xy, x+y}. This is partially based on joint work with Marcin Sabok.

Petr Naryshkin (Munster)

Title: Borel asymptotic dimension and hyperbolic groups

Time: Thursday, 15.02.2024, 16:00.
Place: ELTE
Abstract: We will give a brief overview of the hyperfiniteness question for orbit equivalence relations and define the Borel asymptotic dimension. We will provide an easy proof of the theorem of Marquis and Sabok, which states that the actions of hyperbolic groups of their Gromov boundaries are hyperfinite. If time permits, we will also explain how the same holds for free topologically amenable actions of free groups. The talk is based on joint works with Andrea Vaccaro.

Clark Lyons (UCLA/Berkeley)

Title: Baire Measurable Matchings in Non-amenable Graphs

Time: 08.02.2024, 16:00.
Place: Renyi Institute, Tondo Room
Abstract: Tutte's theorem provides a necessary and sufficient condition for a finite graph to have a perfect matching. In this talk I will present joint work with Kastner showing that if a locally finite Borel graph satisfies a strengthened form of Tutte's condition, then it has a perfect matching which is Baire measurable. As a consequence, the Schreier graph of a free action of a non-amenable group on a Polish space admits a Baire measurable perfect matching. This is analogous to the result of Csoka and Lippner on factor of IID perfect matchings for non-amenable Cayley graphs.

Felix Weilacher (CMU)

Title: Computable vs. Descriptive Combinatorics of Local Problems

Time: 07.02.2024, 14:00.
Place: Renyi Institute, Tondo Room
Abstract: We consider "local" combinatorial problems on graphs. I.e, problems in which we seek a global labelling of vertices, edges, etc. satisfying some set of local constraints. Typical examples include proper coloring and perfect matching. We are moreover interested in finding solutions which are in some sense "constructive" or "definable". We will focus on two specializations of this: finding Baire measurable solutions for Borel graphs on Polish spaces, and finding computable solutions for computable graphs on the natural numbers. Recent investigations have uncovered a large number of similarities between these two settings, but there are interesting questions about how deep the relationship really is. We will attempt to survey some recent positive and negative results in this direction. Includes joint work with Qian, Bowen, and Conley.


09.13. ZV: On a theorem of Bourgain-Fremlin-Talagrand
09.20. ZV: On a theorem of Bourgain-Fremlin-Talagrand
09.27. ZV: On a theorem of Bourgain-Fremlin-Talagrand
10.04. Anett Kocsis: On a note of Weilacher
10.11. Anett Kocsis: On a note of Weilacher
10.18. Balázs Bursics: Borel boundedness
10.25. Balázs Bursics: Borel boundedness
11.08. Máté Pálfy: Cost
11.15. Máté Pálfy: Cost
11.22. Máté Pálfy: Cost
11.28. Donát Pigler: Martin's Conjecture
12.06. Julia Millhouse (Vienna): A minimal counterexample to a uniform strengthening of Ramsey's Theorem
12.13. Donát Pigler: Martin's Conjecture
12.20. Jonathan Schilhan (Leeds): A geometric condition for Dependent Choice