- Extremal graph theory
- Probabilistic combinatorics
- Combinatorial number theory
- Ramsey theory
Andrew’s research interests lie in a range of different areas. In recent years his work has had a particular focus on ‘independent set’ problems in the setting of graphs, posets and the integers. This work has been supported by an EPSRC Fellowship (2015-2018). He also has interests in extremal and probabilistic combinatorics, and Ramsey theory.
A famous result of Green and Sapozhenko determines the number of sum-free sets in the first n natural numbers. In the 1990s, Cameron and Erdős raised the question of how many maximal sum-free sets there are in this setting. Through a graph theoretical approach, Andrew and his co-authors (J. Balogh, H. Liu and M. Sharifzadeh) gave a solution to this important question.
Much of Andrew’s recent work also considers questions in Ramsey theory. Together with his PhD student R. Hancock, and K. Staden, he generalised the random Ramsey theorem of Rödl and Ruciński by providing a resilience analogue. They also resolved a general subcase of the asymmetric random Ramsey conjecture of Kohayakawa and Kreuter.
One of the most central results in Ramsey theory is Goodman’s theorem from 1959 which determines the minimum number of monochromatic triangles in a 2-coloured complete graph. Andrew and his co-authors (J. Cummings, D. Kral, F. Pfender, K. Sperfeld and M. Young) have solved this problem for 3-coloured graphs, thereby solving a classical problem of Goodman.
In the past Andrew has made progress on a number of problems concerning graph decompositions. In particular, together with B. Csaba, D. Kühn, A. Lo and D. Osthus, he has solved the beautiful 1-factorization conjecture for large graphs. This classical conjecture gives a condition for a regular graph to have a decomposition into perfect matchings.
Andrew has also written a number of papers on (hyper)graph embedding problems. For example, in a sequence of several papers, he and his co-authors (D. Kühn, D. Osthus and Y. Zhao) have established a number of minimum