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Finite and Descriptive Combinatorics

This is the webpage ( 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).


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

* = co-advised PhD student



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

  1. L.Grabowski, A.Mathe, OP: Measurable equidecompositions for group actions with an expansion property, accepted by J Eur Math Soc, 48 pages.
  2. 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
  3. 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.
  4. M.Kang, T.Makai and OP: Supersaturation Problem for the Bowtie, European J Comb 88 (2020) 103107.
  5. JG and OP: Measurable versions of Vizing's theorem, Advances in Mathematics, 374 (2020) paper 107378, 40pp
  6. J.Kim, H.Liu, OP and M.Sharifzadeh: Asymptotic Structure for the Clique Density Theorem, Discrete Analysis (2020) Paper 19, 26pp
  7. 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
  8. C.T.Conley, JG and OP: Divisibility of Spheres with Measurable Pieces, 31pp
  9. H.Liu, OP, M.Sharifzadeh and K.Staden: Stability from graph symmetrisation arguments with applications to inducibility, 41pp
  10. JG and I. Rocha: Fractional Isomorphism of Graphons, accepted to Combinatorica, 24pp
  11. JG: Approximate Schreier decorations and approximate Konig line coloring theorem, accepted by AHL, 11pp
  12. JG and Z. Vidnyánszky: Tall Fσ subideals of tall analytic ideals, accepted to PAMS, 5pp
  13. JG and OP: Large Deviation Principles for Block and Step Graphon Random Graph Models, 17pp
  14. O.Cooley, M.Kang and OP: On a question of Vera T. Sos about size forcing of graphons, 22pp
  15. 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
  16. JG and V.Rozhon: Classification of Local Problems on Paths from the Perspective of Descriptive Combinatorics, extended abstract accepted at EUROCOMB2021, 34pp
  17. OP and K.Staden Stability for the Erdos-Rothschild problem, 50pp
  18. OP and K.Staden Exact solutions for the Erdos-Rothschild problem, 53pp
  19. 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, accepted at ITCS, 72pp
  20. I.Gil Fernandez, J.Kim, H.Liu, OP New lower bounds on kissing numbers and spherical codes in high dimensions, 20pp
  21. S. Brandt, Y. Chang, JG, C. Grunau, V. Rozhoň, Z. Vidnyánszky On Homomorphism Graphs, 21pp
  22. V.Falgas-Ravry, OP, E.Vaughan, J.Volec The codegree threshold of K4-, 31 pages
  23. A.Mathe, J.Noel, OP Circle Squaring with Pieces of Small Boundary and Low Borel Complexity, 49 pages
  24. E.Csoka, L.Grabowski, A.Mathe, OP, K.Tyros Moser-Tardos Algorithm with small number of random bits, 31pp.
  25. JG, R. Greenfeld, V. Rozhoň, T. Tao Measurable tilings by abelian group actions, 25pp.
  26. S. Brandt, Y. Chang, JG, C. Grunau, V. Rozhoň, Z. Vidnyánszky Deterministic Distributed algorithms and Descriptive Combinatorics on Δ-regular trees, 47pp.
  27. I.Gil Fernandez, J.Hyde, H.Liu OP, Z.Wu Disjoint isomorphic balanced clique subdivisions, 17pp.
  28. J. Hou, H. Li, XL, D. Mubayi, Y. Zhang Hypergraphs with infinitely many extremal constructionsLink opens in a new window, 30pp
  29. 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

Talks Given/Forthcoming


  • 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


  • 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


  • 11 Jan: OP, Combinatorics Seminar, Hebrew University of Jerusalem
  • 22 Feb: OP, Caltech Logic Seminar
  • 10 March: JG, UBC probability seminar
  • 1 June: JG, Florida Logic seminar
  • 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
  • 6-10 Sep: JG, EUROCOMB 2021
  • 11 Oct: JG, DIMAP seminar, Warwick University
  • 8 Nov: JG, Descriptive Dynamics and Combinatorics Seminar, McGill University
  • 30 Nov: OP, Warwick Maths Society



  • 1-7 Jan: OP, Workshop "Combinatorics", Oberwolfach