OUPPS (Open University People Profile System)

Biography

Professional biography

I am a lecturer in pure maths in the School of Mathematics and Statistics; I joined the OU in January 2022. I was an undergraduate at Trinity Hall, University of Cambridge, and received a PhD from the University of Birmingham in 2015, followed by postdoctoral research at Warwick and Oxford.

Research interests

I am interested in combinatorics (discrete mathematics); in particular extremal and probabilistic combinatorics. Recently I've been working on graph decompositions and some problems in extremal graph theory.

I am a member of the OU combinatorics and algebra group.

My work is supported by an EPSRC postdoctoral fellowship (2022-25).

Publications

All my papers are available on ArXiv.

Submitted papers

'A framework for the generalised Erdős-Rothschild problem and a resolution of the dichromatic triangle case’ (with P. Gupta, Y. Pehova and E. Powierski), 2025, arXiv:2502.12291. (58pp)
'The semi-inducibility problem’ (with A. Basit, B. Granet, D. Horsley and A. Kündgen), 2025, arxiv:2501.09842. (43pp)
'Transversals via regularity II’ (with Y. Cheng), 2024. (19pp)
'Stability of transversal Hamilton cycles and paths’ (with Y. Cheng), 2024, arXiv:2403.09913. (30pp)
'Transversals via regularity’ (with Y. Cheng), 2023, arXiv:2306.03595. (41pp)
'Geometric constructions for Ramsey-Turán theory’ (with H. Liu, C. Reiher, M. Sharifzadeh), 2021, arXiv:2103.10423. (27pp)

Published papers

‘Ringel’s tree-packing conjecture in quasirandom graphs’ (with P. Keevash), 2023, J. European Math. Soc., to appear. (58pp)
'Universality for transversal Hamilton cycles’ (with C. Bowtell, P. Morris and Y. Pehova), 2023, arXiv:2310.04138. Accepted in Bull. London Math. Soc. (19pp)
'Exact results for the Erdős-Rothschild problem’ (with O. Pikhurko), 2023, Forum Math Sigma 12, E8. (45pp)
'Stability for the Erdős-Rothschild problem’ (with O. Pikhurko), 2023, Forum Math., Sigma 11, E23. (43pp)
‘Stability from symmetrisation arguments with applications to inducibility’ (with H. Liu, O. Pikhurko and M. Sharifzadeh), 2023, J. London Math. Soc., 108 (2), 1121-1162.
‘The generalised Oberwolfach problem’ (with P. Keevash), 2022, J. Combin. Theory B 152, pp. 281-318.
'On the maximum number of integer colourings with forbidden monochromatic sums’ (with H. Liu and M. Sharifzadeh), 2021, Elec. J. Combin., 28 (1), P1.59. (35pp)
'The bandwidth theorem for locally dense graphs’ (with A. Treglown), 2020, Forum Math. Sigma 8, E42. (36pp)
‘The exact minimum number of triangles in graphs of given order and size’ (with H. Liu and O. Pikhurko), 2020, Forum Math. Pi 8, E8. (144pp)
'Minimum number of additive tuples in groups of prime order’ (with O. Chervak and O. Pikhurko), 2019, Elec. J. Combin. 26 Paper 1.30. (16pp)
'Independent sets in hypergraphs and Ramsey properties of graphs and the integers’ (with R. Hancock and A. Treglown), 2019, SIAM J. Disc. Math. 33, pp. 153-188.
‘Proof of Komlós’s conjecture on Hamiltonian subsets’ (with J. Kim, H. Liu and M. Sharifzadeh), 2017, Proc. London Math. Soc. 115 (5), pp. 974-1013.
'The Erdős-Rothschild problem on edge-colourings with forbidden monochromatic cliques’ (with O. Pikhurko and Z.B. Yilma), 2017, Math. Proc. Cambridge Phil. Soc. 163, pp. 341-356.
'Local conditions for exponentially many subdivisions’ (with H. Liu and M. Sharifzadeh), 2017, Combin. Probab. Comput. 26 (3), pp. 423-430.
'On degree sequences forcing the square of a Hamilton cycle’ (with A. Treglown), 2017, SIAM J. Disc. Math. 31 (1), pp. 383-437.
'Solution to a problem of Bollobás and Häggkvist on Hamilton cycles in regular graphs’ (with D. Kühn, A. Lo, D. Osthus), 2016, J. Combin. Theory B 121 (Special issue, ‘Fifty Years of the Journal of Combinatorial Theory’), pp. 85-145.
‘The robust component structure of dense regular graphs and applications’ (with D. Kühn, A. Lo, D. Osthus), 2015, Proc. London Math. Soc. 110 (1), pp. 19-56.
'Approximate Hamilton decompositions of robustly expanding regular digraphs’ (with D. Osthus), 2013, SIAM J. Disc. Math., 27 (3), pp. 1372-1409.

Projects

Graph decompositions via probability and designs

The notion of decomposition, or splitting a larger object into smaller pieces, is ubiquitous in mathematics. Sometimes one does this to better understand the larger object, for example representing a function by a Fourier series, or factorising an integer. On the other hand, we may wish to understand which pieces can possibly partition a given larger object. With which shapes can we tile the plane? Is it possible to 'decompose' the computationally expensive operation of division into only addition, subtraction and multiplication (as in an algorithm used by a computer to divide real numbers)? This proposal seeks to investigate these problems in graphs, or networks. A graph is a collection of nodes in which some pairs are joined by edges, to represent some relationship or connection between them. More complicated relationships are encoded by hypergraphs, where more than two nodes can lie in an edge together. Graphs are used to model and describe many different systems in biology, communications and computer science and their theoretical study comes under the mathematical field of combinatorics. In the graph setting, the goal of a decomposition problem is to start with a large 'host' graph with many edges, and a collection of 'guest' graphs each with few edges, and to try to fit the guest graphs perfectly into the host graph, using each edge exactly once. This type of problem is one of the oldest in combinatorics, going back to a 1792 question of Euler. The case where each guest graph contains few nodes is by now fairly well-understood and constitutes the area of design theory. This project investigates the other end of the spectrum where guest graphs may contain a number of nodes comparable with the host graph. An example of such a question that can be expressed in the language of graphs, the recently solved Oberwolfach problem, asks for a sequence of seating plans which allow each person in a group to sit next to each other person exactly once over the course of several meals. Very recent successes in this area, including work in which I was involved, solved a number of longstanding conjectures and overcame the barriers met in previous works by novel application of tools from disciplines outside of combinatorics. I seek to build on these successes and draw from tools in probability and design theory to obtain graph decompositions. For example, an effective strategy has been to use a randomised algorithm that makes successive coin flips to choose where to put each guest edge; tools from probability are needed to analyse its evolution. Such an algorithm is very unlikely to be able to place the final pieces correctly; tools from design theory will help complete the decomposition.