I am a Professor of Discrete Mathematics and a Tutorial Fellow in Wadham College. Until recently, I held a Royal Society
University Research Fellow. Before that, I held my Royal Society University Research Fellow in DPMMS and was a College
Research Associate at St John's College, Cambridge. From 2007 to 2010, I was a Junior Research Fellow at
St John's College. My PhD was completed under the supervision of Professor W.T. Gowers. 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:
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, submitted.
The Ramsey number of books, submitted.
On the extremal number of subdivisions, with J. Lee, submitted.
Tower-type bounds for unavoidable patterns in words, with J. Fox and B. Sudakov, to appear in Trans. Amer. Math. Soc.
Online Ramsey numbers and the Subgraph Query Problem, with J. Fox, A. Grinshpun and X. He, to appear in Lovász 70 volume.
Some advances on Sidorenko's conjecture, with J. H. Kim, C. Lee and J. Lee, to appear in J. London Math. Soc. See here for a companion note.
Lines in Euclidean Ramsey theory, with J. Fox, to appear in Discrete Comput. Geom.
Graphs with few paths of prescribed length between any two vertices, to appear in Bull. Lond. Math. Soc.
Hypergraph cuts above the average, with J. Fox, M. Kwan and B. Sudakov, to appear in Israel J. Math.
Hypergraph expanders from Cayley graphs, to appear in Israel J. Math.
Intervals in the Hales–Jewett theorem, with N. Kamčev, to appear in Eur. J. Combin.
Quasirandomness in hypergraphs, with E. Aigner-Horev, H. Hàn, Y. Person and M. Schacht, Electron. J. Combin. (2018), P3.34.
Rational exponents in extremal graph theory, with B. Bukh, J. Eur. Math. Soc. 20 (2018), 1747-1757.
Hereditary quasirandomness without regularity, with J. Fox and B. Sudakov, Math. Proc. Cam. Phil. Soc. 164 (2018), 385-399.
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.
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.
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.
Short proofs of some extremal results II, with J. Fox and B. Sudakov, J. Combin. Theory Ser. B 121 (2016), 173-196.
Ramsey numbers of cubes versus cliques, with J. Fox, C. Lee and B. Sudakov, Combinatorica 36 (2016), 37-70.
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.
Recent developments in graph Ramsey theory, with J. Fox and B. Sudakov, Surveys in Combinatorics 2015, 49-118.
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.
The Erdős-Gyárfás problem on generalized Ramsey numbers, with J. Fox, C. Lee and B. Sudakov, Proc. Lond. Math. Soc. 110 (2015), 1-18.
Distinct volume subsets, with J. Fox, W. Gasarch, D. G. Harris, D. Ulrich and S. Zbarsky, SIAM J. Discrete Math. 29 (2015), 472-480.
Cycle packing, with J. Fox and B. Sudakov, Random Structures Algorithms 45 (2014), 608-626.
On the KŁR conjecture in random graphs, with W. T. Gowers, W. Samotij and M. Schacht, Israel J. Math. 203 (2014), 535-580.
The Green-Tao theorem: an exposition, with J. Fox and Y. Zhao, EMS Surv. Math. Sci. 1 (2014), 249-282.
Combinatorial theorems relative to a random set, Proceedings of the International Congress of Mathematicians 2014, Vol. 4, 303-328.
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.
Extremal results in sparse pseudorandom graphs, with J. Fox and Y. Zhao, Adv. Math. 256 (2014), 206-290.
Short proofs of some extremal results, with J. Fox and B. Sudakov, Combin. Probab. Comput. 23 (2014), 8-28.
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.
On two problems in graph Ramsey theory, with J. Fox and B. Sudakov, Combinatorica 32 (2012), 513-535.
Bounds for graph regularity and removal lemmas, with J. Fox, Geom. Funct. Anal. 22 (2012), 1191-1256.
Erdős-Hajnal-type theorems in hypergraphs, with J. Fox and B. Sudakov, J. Combin. Theory Ser. B 102 (2012), 1142-1154.
On the Ramsey multiplicity of complete graphs, Combinatorica 32 (2012), 171-186.
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.
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.
Hypergraph Ramsey numbers, with J. Fox and B. Sudakov, J. Amer. Math. Soc. 23 (2010), 247-266.
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.