- Extremal graph theory
- Probabilistic combinatorics
- Ramsey theory
- Graph decompositions
- Hypergraphs and directed graphs
Andrew’s primary research interests lie in extremal graph theory. His most recent work has been on 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 (see http://en.wikipedia.org/wiki/1-factorization_conjecture#1-factorization_conjecture).
This classical conjecture gives a condition for a regular graph to have a decomposition into perfect matchings.
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. Recently, 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.
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 degree conditions that ensure a hypergraph contains a perfect matching.