I am a Professor of Mathematics here at Caltech, having recently moved from Oxford, where
I was a University Lecturer and a Tutorial Fellow at Wadham College. For a full CV, see here.
My principal research interests are in combinatorics and number theory, particularly Ramsey
theory, extremal graph theory, additive combinatorics, pseudorandomness and random graphs.
The notes for a short course in Ramsey theory may be found here.
Notes for a course on extremal graph theory are here.
Notes for a summer school on pseudorandom graphs are here.
Here are some of my papers:
The regularity method for graphs with few 4-cycles, with J. Fox, B. Sudakov and Y. Zhao, submitted.
Repeated patterns in proper colourings, with M. Tyomkyn, submitted.
Ramsey numbers of books and quasirandomness, with J. Fox and Y. Wigderson, submitted.
A new bound for the Brown–Erdős–Sós problem, with L. Gishboliner, Y. Levanzov and A. Shapira, submitted.
Short proofs of some extremal results III, with J. Fox and B. Sudakov, submitted.
Monochromatic combinatorial lines of length three, submitted.
Hypergraph expanders of all uniformities from Cayley graphs, with J. Tidor and Y. Zhao, submitted.
Sidorenko's conjecture for blow-ups, with J. Lee, to appear in Discrete Anal.
Ramsey games near the critical threshold, with S. Das, J. Lee and T. Mészáros, to appear in Random Structures Algorithms.
More on the extremal number of subdivisions, with O. Janzer and J. Lee, to appear in Combinatorica.
On the extremal number of subdivisions, with J. Lee, to appear in Int. Math. Res. Not.
Intervals in the Hales–Jewett theorem, with N. Kamčev, to appear in Eur. J. Combin.
Books versus triangles at the extremal density, with J. Fox and B. Sudakov, SIAM J. Discrete Math. 34 (2020), 385-398.
The Ramsey number of books, Adv. Combin. 2019, Paper No. 3, 12 pp.
Graphs with few paths of prescribed length between any two vertices, Bull. Lond. Math. Soc. 51 (2019), 1015-1021.
Online Ramsey numbers and the Subgraph Query Problem, with J. Fox, A. Grinshpun and X. He, in Building Bridges II, 159-194, Springer, 2019.
Hypergraph cuts above the average, with J. Fox, M. Kwan and B. Sudakov, Israel J. Math. 233 (2019), 67-111.
Hypergraph expanders from Cayley graphs, Israel J. Math. 233 (2019), 49-65.
Lines in Euclidean Ramsey theory, with J. Fox, Discrete Comput. Geom. 61 (2019), 218-225.
Tower-type bounds for unavoidable patterns in words, with J. Fox and B. Sudakov, Trans. Amer. Math. Soc. 372 (2019), 6213-6229.
Rational exponents in extremal graph theory, with B. Bukh, J. Eur. Math. Soc. 20 (2018), 1747-1757.
Some advances on Sidorenko's conjecture, with J. H. Kim, C. Lee and J. Lee, J. London Math. Soc. 98 (2018), 593-608. See here for a companion note.
Quasirandomness in hypergraphs, with E. Aigner-Horev, H. Hàn, Y. Person and M. Schacht, Electron. J. Combin. (2018), P3.34.
Hereditary quasirandomness without regularity, with J. Fox and B. Sudakov, Math. Proc. Cam. Phil. Soc. 164 (2018), 385-399.
Finite reflection groups and graph norms, with J. Lee, Adv. Math. 315 (2017), 130-165.
Hedgehogs are not colour blind, with J. Fox and V. Rödl, J. Combin. 8 (2017), 475-485.
Freiman homomorphisms on sparse random sets, with W. T. Gowers, Q. J. Math. 68 (2017), 275-300.
Almost-spanning universality in random graphs, with A. Ferber, R. Nenadov and N. Škorić, Random Structures Algorithms 50 (2017), 380-393.
Quasirandom Cayley graphs, with Y. Zhao, Discrete Anal. 2017, Paper No. 6, 14 pp.
A sequence of triangle-free pseudorandom graphs, Combin. Probab. Comput. 26 (2017), 195-200.
A note on induced Ramsey numbers, with D. Dellamonica Jr., S. La Fleur, V. Rödl and M. Schacht, in A Journey Through Discrete Mathematics, 357-366, Springer, 2017.
Ordered Ramsey numbers, with J. Fox, C. Lee and B. Sudakov, J. Combin. Theory Ser. B. 122 (2017), 353-383.
Combinatorial theorems in sparse random sets, with W. T. Gowers, Ann. of Math. 184 (2016), 367-454.
Ramsey numbers of cubes versus cliques, with J. Fox, C. Lee and B. Sudakov, Combinatorica 36 (2016), 37-70.
Short proofs of some extremal results II, with J. Fox and B. Sudakov, J. Combin. Theory Ser. B 121 (2016), 173-196.
Monochromatic cycle partitions in local edge colourings, with M. Stein, J. Graph Theory 81 (2016), 134-145.
A relative Szemerédi theorem, with J. Fox and Y. Zhao, Geom. Funct. Anal. 25 (2015), 733-762. See here for a companion note.
The Erdős–Gyárfás problem on generalized Ramsey numbers, with J. Fox, C. Lee and B. Sudakov, Proc. Lond. Math. Soc. 110 (2015), 1-18.
On the grid Ramsey problem and related questions, with J. Fox, C. Lee and B. Sudakov, Int. Math. Res. Not. 17 (2015), 8052-8084. See here for a companion note.
Recent developments in graph Ramsey theory, with J. Fox and B. Sudakov, Surveys in Combinatorics 2015, 49-118.
Distinct volume subsets, with J. Fox, W. Gasarch, D. G. Harris, D. Ulrich and S. Zbarsky, SIAM J. Discrete Math. 29 (2015), 472-480.
Combinatorial theorems relative to a random set, Proceedings of the International Congress of Mathematicians 2014, Vol. 4, 303-328.
On the KŁR conjecture in random graphs, with W. T. Gowers, W. Samotij and M. Schacht, Israel J. Math. 203 (2014), 535-580.
Extremal results in sparse pseudorandom graphs, with J. Fox and Y. Zhao, Adv. Math. 256 (2014), 206-290.
The Green–Tao theorem: an exposition, with J. Fox and Y. Zhao, EMS Surv. Math. Sci. 1 (2014), 249-282.
Cycle packing, with J. Fox and B. Sudakov, Random Structures Algorithms 45 (2014), 608-626.
Short proofs of some extremal results, with J. Fox and B. Sudakov, Combin. Probab. Comput. 23 (2014), 8-28.
Ramsey-type results for semi-algebraic relations, with J. Fox, J. Pach, B. Sudakov and A. Suk, Trans. Amer. Math. Soc. 366 (2014), 5043-5065.
A preliminary version of this paper appeared in the Proceedings of the Twenty-ninth Annual Symposium on Computational Geometry, SoCG '13, Rio de Janeiro, Brazil, 309-318.
Two extensions of Ramsey's theorem, with J. Fox and B. Sudakov, Duke Math. J. 162 (2013), 2903-2927.
Graph removal lemmas, with J. Fox, Surveys in Combinatorics 2013, 1-50.
The Ramsey number of dense graphs, Bull. Lond. Math. Soc. 45 (2013), 483-496.
An improved bound for the stepping-up lemma, with J. Fox and B. Sudakov, Discrete Appl. Math. 161 (2013), 1191-1196.
Bounds for graph regularity and removal lemmas, with J. Fox, Geom. Funct. Anal. 22 (2012), 1191-1256.
On two problems in graph Ramsey theory, with J. Fox and B. Sudakov, Combinatorica 32 (2012), 513-535.
On the Ramsey multiplicity of complete graphs, Combinatorica 32 (2012), 171-186.
Erdős–Hajnal-type theorems in hypergraphs, with J. Fox and B. Sudakov, J. Combin. Theory Ser. B 102 (2012), 1142-1154.
Weak quasi-randomness for uniform hypergraphs, with H. Hàn, Y. Person and M. Schacht, Random Structures Algorithms 40 (2012), 1-38.
Large almost monochromatic subsets in hypergraphs, with J. Fox and B. Sudakov, Israel J. Math. 181 (2011), 423-432.
Hypergraph Ramsey numbers, with J. Fox and B. Sudakov, J. Amer. Math. Soc. 23 (2010), 247-266.
An approximate version of Sidorenko's conjecture, with J. Fox and B. Sudakov, Geom. Funct. Anal. 20 (2010), 1354-1366.
An extremal theorem in the hypercube, Electron. J. Combin. 17 (2010), R111. Erratum to this paper.
A new upper bound for diagonal Ramsey numbers, Ann. of Math. 170 (2009), 941-960.
Ramsey numbers of sparse hypergraphs, with J. Fox and B. Sudakov, Random Structures Algorithms 35 (2009), 1-14.
On-line Ramsey numbers, SIAM J. Discrete Math. 23 (2009), 1954-1963.
Hypergraph packing and sparse bipartite Ramsey numbers, Combin. Probab. Comput. 18 (2009), 913-923.
A new upper bound for the bipartite Ramsey problem, J. Graph Theory 58 (2008), 351-356.
On the existence of rainbow 4-term arithmetic progression, with V. Jungić and R. Radoičić, Graphs Combin. 23 (2007), 249-254.
Rainbow solutions of linear equations over Zp, Discrete Math. 306 (2006), 2056-2063.