# Finite and Descriptive Combinatorics

This is the webpage (warwick.ac.uk/meascombLink opens in a new window) of the research group "Finite and Descriptive Combinatorics" whose activities are, in particular, supported by the Leverhulme Research Project Grant RPG-2018-424 "Measurable Combinatorics" (1 October 2019 - 30 September 2023) and the ERC Advanced Grant 101020255 "Finite and Descriptive Combinatorics" (1 January 2022 - 31 December 2026).

## Overview

In addition to the classical directions of extremal and probabilistic combinatorics (Turan function, Ramsey theory, etc), the group also explores emerging deep connections between combinatorics and other fields such as analysis, descriptive set 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.

## Group Members

- Irene Gil Fernandez (since 10/2020) *
- Jan Grebik (since 10/2019)
- Joseph Hyde (1/4/2022 - 31/7/2022)
- Xizhi LiuLink opens in a new window (since 1/4/2022)
- Matteo Mazzamurro (02/2022-01/2023) *
- Oleg Pikhurko (PI)
- Shumin Sun (since 10/2022) *
- Zhuo Wu (since 10/2021) *

* = co-advised PhD student

## News

- 24-28 Jun'24: JG, Alexander Kechris, OP, Stevo Todorcevic, and Zoltán Vidnyánszky organise Workshop on Measurable Combinatorics, Erdos Center, Budapest
- 20-24 Mar'23: Damien Gaboriau, Andrew Marks, OP and Anush Tserunyan organise Workshop on Borel and Measurable Combinatorics in Algebra and Dynamics, Fields Institute, Toronto
- Feb-Mar'23: OP serves on the Program Committee of Digital Theme UK-Ukraine Research Twinning Conference
- Jan'23- : OP teaches at the weekly Math Challenge Club at St Augustine's Catholic Primary School, Kenilworth
- 26-27 Jan'23: OP serves on the Mathematics Review Panel of the Academy of Finland
- 18 Jan'23: Matteo Mazzamurro successfully defends his PhD viva. Congratulations, Matteo!!!
- Dec'22-Feb'23: OP serves on the Selection Committee of the MSCA4Ukraine programme
- Dec'22: Discovery magazine publishes an article about group's work on circle squaring
- Oct'22: JG wins the 2022 Mary Ellen Rudin Young Researcher Award. Congratulations, Jan!!!
- 14 Jun'22: JH organises Birmingham-Warwick One-Day Combinatorics Meeting
- Jun-Jul'22: OP serves on the hiring panel for the Professor Post in Discrete Maths and Computer Science at Warwick
- May'22- : OP serves on the committee of INI's programme "Solidarity for mathematicians: refugees of war"
- May'22: Group's work on circle squaring is mentioned in "Mathematics News Flash" of LMS Newsletter
- Mar'22: El Periodico publishes an article about group's work on circle squaring
- 10 Mar'22: OP serves on PhD committee of Xizhi Liu, who successfully defended his thesis "Extremal Hypergraph Problems". Congratulations, Xizhi!
- Feb'22: OP serves as reviewer of membership applications for IMTech
- Feb'22: Quanta publishes an article about group's work on circle squaring
- 13-19 Feb'22: JG, OP and Anush Tserunyan organise Oberwolfach Mini-Workshop "Descriptive Combinatorics, LOCAL Algorithms and Random Processes"
- 3-4 Feb'22: OP serves on the Mathematics Review Panel of the Academy of Finland
- Jan'22: Some results co-authored by grouthere isp members are mentioned in Bogdan Grechuk's new book "Landscape of 21st Century Mathematics: Selected Advances, 2001-2020"
- Dec'21: OP serves as expert evaluator for Postdoctoral Junior Leader Fellowships of the “la Caixa” Foundation
- 12 Oct - 7 Dec'21: JG lectures mini-course "Distributed computing, random processes and Borel combinatorics"
- JG and OP organise 2021-22 Warwick's Combinatorics Seminar
- 2-7 Aug'21: OP is a team leader of Warwick's team at the 28th International Mathematics Competition
- 14 Jul'21: OP was the internal examiner of Bogdan Alecu's PhD thesis that was successfully defended. Congratulations, Bogdan!
- Jul'21: JG was awarded Warwick's Faculty of Science Postdoctoral Prize. Congratulations, Jan!
- Apr'21: OP was awarded a 5-year ERC Advanced Grant for €1.58m (see also Warwick News, Ukraine Network)
- Dec'20-Aug'21: OP serves at the Program Committee of Eurocomb 2021
- 1 Dec'20: OP was the internal examiner of Yani Pehova's PhD thesis that was successfully defended. Congratulations, Yani!
- Nov'20: OP appointed to the Scientific Advisory Board of IMTech
- JG and OP organise 2020-21 Warwick's Combinatorics Seminar
- 25-30 Jul'20: George Kontogeorgiou and OP are team leaders of Warwick's team at the 27th International Mathematics Competition
- 2 Jul'20: OP was an external examiner of Jan Corsten's PhD thesis at LSE, that was successfully defended. Congratulations, Jan!
- 5 May'20: Jan Grebik successfully defended his PhD thesis "Definable graphs". Congratulations!
- 1 Jan'20: OP appointed to the Editorial Board of "Random Structures and Algorithms"
- 4 Nov'19: OP was an external examiner of François Pirot's PhD thesis at Radboud University, Nijmegen, that was successfully defended. Congratulations, François!

## 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, accepted by J Eur Math Soc, 48 pages.
- A.Blumenthal, B.Lidicky, Y.Pehova, F.Pfender, OP and J.Volec: Sharp bounds for decomposing graphs into edges and triangles, Combin Prob Comput, 30 (2021) 271-287
- 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, Advances in Mathematics, 374 (2020) paper 107378, 40pp
- J.Kim, H.Liu, OP and M.Sharifzadeh: Asymptotic Structure for the Clique Density Theorem, Discrete Analysis (2020) Paper 19, 26pp
- OP: Borel Combinatorics of Locally Finite Graphs, in "Surveys in Combinatorics 2021" (edited by K.K.Dabrowski et al), the invited volume of the 28th British Combinatorial Conference, pages 267-319
- C.T.Conley, JG and OP: Divisibility of Spheres with Measurable Pieces, accepted by L'Enseignement Mathematique, 34pp.
- H.Liu, OP, M.Sharifzadeh and K.Staden: Stability from graph symmetrisation arguments with applications to inducibility, 41pp
- JG and I. Rocha: Fractional Isomorphism of Graphons, Combinatorica, 42 (2022) 365–404
- JG: Approximate Schreier decorations and approximate Konig line coloring theorem, Annales Henri Lebesgue, 5 (2022) 303-315.
- JG and Z. Vidnyánszky: Tall F
_{σ}subideals of tall analytic ideals, accepted to PAMS, 5pp - JG and OP: Large Deviation Principles for Block and Step Graphon Random Graph Models, 17pp
- O.Cooley, M.Kang and OP: On a question of Vera T. Sos about size forcing of graphons, accepted by Acta Mathematica Hungarica, 22pp
- JG and Vasek Rozhon: Local Problems on Grids from the Perspective of Distributed Algorithms, Finitary Factors, and Descriptive Combinatorics (previous, now irrelevant, title: Of Toasts and Tails), 48pp
- JG and V.Rozhon: Classification of Local Problems on Paths from the Perspective of Descriptive Combinatorics, extended abstract accepted at EUROCOMB2021, 34pp
- OP and K.Staden Stability for the Erdos-Rothschild problem, 50pp
- OP and K.Staden Exact solutions for the Erdos-Rothschild problem, 53pp
- S. Brandt, Y. Chang, JG, C. Grunau, V. Rozhoň, Z. Vidnyánszky Local Problems on Trees from the Perspectives of Distributed Algorithms, Finitary Factors, and Descriptive Combinatorics, 13th Innovations in Theoretical Computer Science Conference, Art. No. 29, 26 pp., LIPIcs. Leibniz Int. Proc. Inform., 215, Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern, 2022.
- IGF, J.Kim, H.Liu, OP New lower bounds on kissing numbers and spherical codes in high dimensions, 20pp
- S. Brandt, Y. Chang, JG, C. Grunau, V. Rozhoň, Z. Vidnyánszky On Homomorphism Graphs, 21pp
- V.Falgas-Ravry, OP, E.Vaughan, J.Volec The codegree threshold of K4-, 31 pages
- A.Mathe, J.Noel, OP Circle Squaring with Pieces of Small Boundary and Low Borel Complexity, 49 pages
- E.Csoka, L.Grabowski, A.Mathe, OP, K.Tyros Moser-Tardos Algorithm with small number of random bits, 31pp.
- JG, R. Greenfeld, V. Rozhoň, T. Tao Measurable tilings by abelian group actions, accepted by IMRN, 25pp.
- S. Brandt, Y. Chang, JG, C. Grunau, V. Rozhoň, Z. Vidnyánszky Deterministic Distributed algorithms and Descriptive Combinatorics on Δ-regular trees, 47pp.
- IGF, JH, H.Liu OP, ZW Disjoint isomorphic balanced clique subdivisions, 17pp.
- J. Hou, H. Li, XL, D. Mubayi, Y. Zhang Hypergraphs with infinitely many extremal constructionsLink opens in a new window, 30pp
- XL, S. Mukherjee Tight query complexity bounds for learning graph partitionsLink opens in a new window, 35th Annual Conference of Learning Theory (COLT), 2022, 12pp
- JH Towards the 0-statement of the Kohayakawa-Kreuter conjecture, accepted to Combinatorics, Probability and Computing, 39pp
- XL and OP Hypergraph Turan densities can have arbitrarily large algebraic degree, 9pp.
- S.Cambie, Y.Dong and MM Extremal values of degree-based entropies of bipartite graphs, 13pp
- S.Cambie and MM Resolution of Yan's conjecture on entropy of graphs, accepted by MATCH, 9pp
- S.Cambie and MM Extremal entropy for graphs with given size, 8pp.
- IGF and H.Liu How to build a pillar: a proof of Thomassen's conjecture, 16pp
- IGF, J.Kim, Y.Kim, H.Liu Nested cycles with no geometric crossings, to appear in Proceedings of the American Mathematical Society, 10pp
- JG, Z. Vidnyánszky Ramsey, expanders, and Borel chromatic numbers, 11pp
- (with S.Glock, F.Joos, J.Kim, M.Kuhn, L.Lichev) On the (6,4)-problem of Brown, Erdos and Sos, 11pp
- OP and XL Finite Hypergraph Families with Rich Extremal Turan Constructions via Mixing Patterns, 57pp
- ZW Positive co-degree Turan number for C5 and C5- , 8 pages
- OP On the limit of the positive l-degree Turan problem, 14pp
- J. Hou, H. Li, XL, L.-T. Yuan, and Y. Zhang A step towards a general density Corradi–Hajnal TheoremLink opens in a new window, 32pp
- A.Grzesik, D.Kral and OP Forcing Generalized Quasirandom Graphs Efficiently, 20pp.
- JG Measurable Vizing's theorem, 25pp.

## 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
- 21 Oct: JG, Probability seminar (online), UBC
- 10 Nov: OP, UIUC Graph Theory and Combinatorics Seminar
- 23 Nov: JG, Extremal and Probabilistic Combinatorics Webinar
- 27 Nov: OP, TU Graz Combinatorics and Optimization Seminar
- 18 Dec: OP, Combinatorics Seminar of Shandong University

2021

- 11 Jan: OP, Combinatorics Seminar, Hebrew University of Jerusalem
- 22 Feb: OP, Caltech Logic Seminar
- 10 Mar: JG, UBC probability seminar
- 5 May: IGF, LIMDA Seminar, Universitat Politècnica de Catalunya,
*Spain* - 19 May: IGF, Warwick Postgraduate Seminar
- 1 Jun: JG, Florida Logic seminar
- 7 Jun: IGF, (Online) Young Researchers in Mathematics Conference, UK
- 21 Jun: OP, Colloquium Discrete Mathematics and Probability, TU-Graz
- 23 Jun: OP, Mini-symposium on Extremal and Probabilistic Combinatorics, 8th European Congress of Mathematics
- 5-9 Jul: OP, the Richard Rado Lecture at the 28th British Combinatorial Conference
- 26 Jul: IGF, Summer school "Matemáticas: mucho más que números"
*,*Universidad de Santiago de Compostela, Spain - 6-10 Sep: JG, EUROCOMB 2021, Barcelona
- 7 Sep: IGF, EUROCOMB 2021, Barcelona
- 11 Oct: JG, DIMAP seminar, Warwick University
- 14 Oct: IGF, Discrete Seminar, Umeå University, Sweden
- 26 Oct: IGF, First Workshop on Developments in Combinatorics, Shandong University, China
- 8 Nov: JG, Descriptive Dynamics and Combinatorics Seminar, McGill University
- 12 Nov: IGF, PhD Seminar on Combinatorics, Games and Optimisation, London School of Economics
- 30 Nov: OP, Warwick Maths Society

2022

- 12 Jan: JG, Combinatorial group seminar, Institute of Computer Science of the Czech Academy of Sciences, Prague
- 7-11 Feb: OP, Workhop "Logic and Random Discrete Structures", Dagstuhl
- 16 Feb: IGF, PG Forum, Lancaster University
- 3 Mar: JG, Algorithms Seminar, Oxford
- 9 Mar: IGF, Warwick Postgraduate Seminar
- 10 Mar: IGF, Combinatorics Seminar, University of Birmingham
- 4-6 Apr: OP, IBS ECOPRO Opening Conference, Daejeon, Korea
- 27 Apr: XL, Warwick Combinatorics Seminar
- 16-20 May: OP, Workshop "Graphs, Groups, Stochastic Processes", Budapest
- 27 Jun - 1 Jul: OP, Third Southwestern German Workshop on Graph Theory, Heildelberg
- 27 Jun - 2 Jul: JG, Descriptive graph theory, San Jose
- 29 Jun: XL, IBS Virtual Discrete Math Colloquium, Korea.
- 4 Jul: IGF, Discrete Mathematics Days 2022, Universidad de Cantabria, Spain
- 8 Jul: XL, The Second Armenian Workshop On Graphs, Combinatorics, Probability, and their application to machine learning, Armenia.
- 20 Jul: XL, Seminar, Data Science Institute, Shangdong University, Jinan, China.
- 8-12 Aug: OP, Conference on Recent Advances in Probabilistic and Extremal Combinatorics, Ascona, Switzerland
- 16 Aug: JG, Workshop on Random Graphs, Combinatorial Limits, Stochastic Processes, Budapest, Hungary
- 18-24 Sep: OP, 9th Polish Combinatorial Conference, Bedlewo
- 28 Oct: JG, Nankai Logic Colloquium (online), Nankai University, China

2023

- 1-7 Jan: OP, Workshop "Combinatorics", Oberwolfach
- 3 Feb: SS, Combinatorics Seminar, Warwick
- 8 Feb: IGF, VI Congreso de jóvenes investigadores de la RSME, Universidad de León, Spain
- 28 Feb: OP, Seminar in Discrete Mathematics, Tsinghua University, Beijing
- 5 Mar: JG, SEALS 2023, University of Florida, Florida
- 6 Mar, JG, Colloquium, University of Florida, Florida
- 7 Mar, JG, Logic seminar, University of Florida, Florida
- 13-17 Mar: OP, Measure Group Theory workshop,Link opens in a new window Centre de Recherches Mathématiques (CRM), Montréal
- 15-18 Mar: JG, 56th Spring Topology and Dynamical Systems Conference (STDC), online,
- 20-24 Mar: JG, Workshop on Borel and Measurable Combinatorics in Algebra and Dynamics, Fields Institute, Toronto
- 26-30 Mar: OP, UA DIGITAL 2023, online
- 27-29 Mar: ZW, Postgraduate Combinatorial Conference, Birmingham
- 17 Mar: XL, Seminar, School of Mathematical Sciences, East China Normal University (ECNU), Shanghai, China.
- 20 Mar: XL, Seminar, Taiyuan University of Technology, Taiyuan, China, online.
- 16-17 Apr: XL, Seventh Workshop on Graph Theory and Combinatorics, Shanghai Jiao Tong University (SJTU), Shanghai, China.
- 19 Apr: XL, SCMS Combinatorics Seminar, Shanghai, China.
- 3 May: JG, Caltech logic seminar, Caltech ,US.
- 11 May: ZW, Postgraduate Seminar, Warwick
- 22-26 May: JG, Measured Group Theory, Stochastic Processes on Groups and Borel Combinatorics, CIRM, Luminy
- 28-31 May: OP, Conference "Spherical packings, coverings and spherical codes" (SPCSC 2023), Sofia
- 17-21 July: JG, The 37th Summer Conference on Topology and Its Applications, Youngstown, Ohio
- 21-25 Aug: OP, Conference "Descriptive set theory & dynamics", Banach Centre, Warsaw
- 28 Aug - 1 Sep: OP, Eurocomb 2023, Prague

## Research visitors

- 9-12 November 2021: Łukasz Grabowski (Lancaster University)
- 10-12 May 2022: Péter Pál Pach (Budapest University of Technology and Economics)
- 16-22 October 2022: Dan Kral' (Masaryk University)
- 16-22 October 2022: Daniel Ilkovic (Masaryk University)
- 15-20 January 2023: Jan Volec (Technical University Prague)
- 4-18 February 2023: Clinton Conley (Carnegie Mellon University)
- 18-21 February 2023: Mihyun Kang (TU-Graz)