# Measurable Combinatorics

This is the webpage of the Leverhulme Research Project Grant RPG-2018-424 "Measurable Combinatorics", 1 October 2019 - 30 September 2023 (warwick.ac.uk/meascomb).

## Overview

This project will explore emerging deep connections between combinatorics and other fields such as analysis, descriptive set theory, ergodic theory, measured group theory, etc, with applications going both ways. One way of applying analytic techniques to finite graphs will be by means of graph limits (analytic objects of bounded complexity that capture asymptotic properties of large graphs). In the other direction, combinatorial techniques will be applied in search of "constructive'' solutions to some foundational mathematical problems, in particular building upon the recent remarkable results that one can split a 2-dimensional disk into ``definable'' pieces and re-arrange them to form a square.

## Current Group Members

- Jan Grebik (1/10/2019-present)
- Oleg Pikhurko (PI)

## Papers

All papers (pre-publication versions) should be freely available from arxiv.org. Please contact one of the authors if you have difficulty accessing them.

- (L.Grabowski, A.Mathe, OP) Measurable equidecompositions for group actions with an expansion property, 46 pages.
- (A.Blumenthal, B.Lidicky, Y.Pehova, F.Pfender, OP and J.Volec) Sharp bounds for decomposing graphs into edges and triangles, accepted by Combin Prob Comput, 20pp.
- (H.Liu, OP and K.Staden) The exact minimum number of triangles in graphs of given order and size, Forum of Math, Pi 8 (2020) 144pp.
- (M.Kang, T.Makai and OP) Supersaturation Problem for the Bowtie, European J Comb 88 (2020) 103107.
- JG and OP Measurable versions of Vizing's theorem, accepted by Advances in Mathematics, 36pp
- OP Borel Combinatorics of Locally Finite Graphs, 51pp

## Talks Given/Forthcoming

2019

- 11 Oct: OP, Workshop "Measurable, Borel, and Topological Dynamics", CIRM
- 4 Nov: OP, Workshop "Structure, Sparsity and Randomness", Radbound University, Nijmegen
- 6 Nov: OP, Old Codger's One-Day Combinatorics Colloquium, Reading
- 21 Nov: OP, 3in1 Workshop on Graph Theory, Doslonce, Poland
- 18 Dec: JG, Seminar on Reckoning, Institute of Mathematics of the Czech Academy of Sciences, Prague
- 20 Dec: JG, Combinatorial group seminar, Institute of Computer Science of the Czech Academy of Sciences, Prague

2020

- 7 Jan: JG, DIMAP seminar, Warwick University
- 14 Jan: OP, Warwick Maths Society
- 30 Jan: OP, ACO Seminar, Carnegie Mellon University
- 11 Feb: OP, seminar, Adam Mickiewicz University, Poznan
- 11 Feb: JG, STUK5 mini-talk, Royal Society building, London
- 17 Feb: OP, Krakow Combinatorics Seminar
- 11 Mar: JG, Algebra and Geometry Seminar, Lancaster University
- 20 May: JG, Caltech logic (online) seminar, Caltech
- 27 Jul: OP, Extremal and Probabilistic Combinatorics Webinar
- 4 & 7 Aug: JG, Midsummer Combinatorial Workshop XXIV, Charles University, Prague
- 1-6 Nov: OP, Dagstuhl Seminar on 'Logic and Random Discrete Structures'
- 10 Nov: OP, UIUC Graph Theory and Combinatorics Seminar
- 27 Nov:, OP, TU Graz Combinatorics and Optimization Seminar

2021

- 5-9 Jul: OP, plenary talk at the 28th British Combinatorial Conference

### Some News

- 4 Nov'19: OP was an external examiner of François Pirot's PhD thesis at Radboud University, Nijmegen, that was successfully defended. Congratulations, François!
- 1 Jan'20: OP appointed to the Editorial Board of "Random Structures and Algorithms"
- 5 May'20: Jan Grebik successfully defended his PhD thesis "Definable graphs". Congratulations!!
- 2 Jul'20: OP was an external examiner of Jan Corsten's PhD thesis at LSE, that was successfully defended. Congratulations, Jan!