To date, the project has led to substantial progress on questions in Ramsey theory, extremal graph theory, additive combinatorics, pseudorandomness and the study of high-dimensional expanders. The main results are summarised below:
- A question of Lovasz and Hatami asks for a characterisation of those graphs which can be used to define norms. Together with the PDRA Joonkyung Lee, the PI uncovered a connection between this question and the theory of finite reflection groups, allowing them to find a wide family of norming graphs. These results also have applications to the study of quasirandomness in hypergraphs and to a famous conjecture of Sidorenko and Erdos-Simonovits. The paper 'Finite reflection groups and graph norms' has now appeared in Advances in Mathematics.
- As a further application of the results on norming graphs, Conlon and Lee made further significant progress on Sidorenko's conjecture, which says that quasirandom graphs minimise the number of copies of any fixed bipartite graph H, where the minimum is taken over all graphs with a given order and density. The main result of this paper states that for any fixed H, there is an appropriate blow-up of H such that the conjecture holds for this blow-up. The resulting paper, 'Sidorenko's conjecture for blow-ups', has been submitted for publication.
- Inspired by the connection between C_6-free graphs and triangle-free pseudorandom graphs, the PDRA Lee and the PI explored the extremal number of the subdivision H_t of the complete graph K_t. Their main result, improving a result of Furedi and answering a problem of Erdos, says that any graph with n vertices and more than C n^{3/2 - c} edges contains a copy of H_t. This paper 'On the extremal number of subdivisions' also raised a large number of interesting open questions that have attracted strong attention.
- Also on extremal numbers is 'Rational exponents in extremal graph theory', written by Boris Bukh and the PI. Here, they build on Bukh's random algebraic method to prove a variant of an old conjecture of Erdos and Simonovits, namely, that for any rational number 1 < r < 2, there exists a family of graphs H_r such that the largest graph on n vertices containing no graph from H_r has between c n^r and C n^r edges. This paper has appeared in the Journal of the European Mathematical Society.
- These papers were followed by another paper with PDRA Lee and Oliver Janzer, 'More on the extremal number of subdivisions'. Here we pursued several themes, all related to the so-called degeneracy problem in extremal graph theory - it is often easy to find homomorphic copies of a graph H, but finding a non-degenerate copy can pose severe difficulties. Our work finds means of circumventing this difficulty, with several applications to problems in the literature.
- In recent years, much energy has gone into the study of high-dimensional, or hypergraph, expanders, with much of the work centring around the study of Ramanujan complexes, remarkable algebraically-defined objects. In 'Hypergraph expanders from Cayley graphs', the PI gives a considerably simpler construction of 3-uniform hypergraphs sharing many of the features of Ramanujan complexes. This paper was later extended to all uniformities by the PI, together with Jonathan Tidor and Yufei Zhao, using additional ideas from coding theory.
- Answering a question of Erdos, Graham, Montgomery, Rothschild, Spencer, and Straus, the PI and Jacob Fox showed that there is a red/blue-colouring of the points of n-dimensional space containing no pair of red points at distance 1 and no 2^{cn} blue points all in a line. This is best possible up to the value of the constant c. The resulting paper, 'Lines in Euclidean Ramsey theory', is due to appear in Discrete and Computational Geometry.
- In 'The Ramsey number of books', the PI solves old problems of Thomason and of Erdos, Faudree, Rousseau and Schelp about the Ramsey number of books, showing that a simple lower bound coming from the probabilistic method can be matched by an upper bound. This problem is intimately connected with other important questions in Ramsey theory, such as estimating diagonal Ramsey numbers and determining the multiplicity of monochromatic cliques. There is also a follow-up, with Jacob Fox and Yuval Wigderson, giving a simpler proof and showing that the extremal configurations are quasirandom.
- Another theme that has drawn much attention in the project is a classical problem of Brown, Erdos and Sos at the interface of Ramsey theory and additive combinatorics. This problem asks for the maximum number of edges in a 3-uniform hypergraph on n vertices with the property that there is no subgraph with v vertices and at least e edges. There have been three papers on the project relating to this important theme. The first, by PDRA Mykhaylo Tyomkyn with Rajko Nenadov and Benny Sudakov, solves the problem when it is known that the hypergraph comes from a finite group. The second, by Tyomkyn and Asaf Shapira, makes substantial progress towards a Ramsey variant of the Brown-Erdos-Sos conjecture posed by the PI. Finally, together with Lior Gishboliner, Yevgeny Levanzov and Asaf Shapira, the PI made substantial progress on the Brown-Erdos-Sos problem itself, improving a result of Sarkozy and Selkow.
- Finally, some of the most striking work in the project was made by PDRA Annika Heckel in her work on the chromatic number of random graphs. For instance, she proved that a simple variant of the chromatic number, the equitable chromatic number, is very tightly concentrated around its expectation. On the other hand, she answered a question originating with Erdos, but repeated often over the last forty years, by showing that the chromatic number itself is not tightly concentrated around its expectation.
