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Combinatorics of the Discrete Hypercube

School of Mathematical Sciences

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Dr JR Johnson Applications accepted all year round

About the Project

The School of Mathematical Sciences of Queen Mary University of London invite applications for a PhD project commencing in September 2020 for self-funded students.

This project will be supervised by Dr Robert Johnson.

The discrete hypercube Q_n has vertex set {0,1\}^n, with two vertices joined by an edge if they differ in a single coordinate. It can also be thought of as the covering relation of the partial order on subsets of an n-element set given by containment, or as the n-dimensional analogue of the usual geometric cube.

Despite its simple to describe construction, the hypercube has a surprisingly complicated structure and there are many tantalising open questions about it. Many questions in extremal set theory (see [B] for instance) are naturally expressed in terms of the hypercube. Graph theoretic concepts such as isoperimetric inequalities and Ramsey results often have interesting analogues in the context of the hypercube. Properties of the hypercube have found application in social choice, computer science and coding theory, often leading to further mathematical problems.

There are numerous possible directions that a PhD project in this area could take. Here are a few areas (one extremal, one structural, and one probabilistic) to give a flavour of the kind of possibilities with a representative problem in each. The exact direction would depend on the background and interests of the candidate.

For full details of this project, please see:

The application procedure is described on the School website. For further inquiries please contact Dr Robert Johnson at [Email Address Removed].

Funding Notes

This project can be undertaken as a self-funded project. Self-funded applications are accepted year-round for a January, April or September start.

The School of Mathematical Sciences is committed to the equality of opportunities and to advancing women’s careers. As holders of a Bronze Athena SWAN award we offer family friendly benefits and support part-time study.


B. Bollobas, Combinatorics: Set Systems, Hypergraphs, Families of Vectors, and Combinatorial Probability, (1985), CUP.

J.R. Johnson and T. Pinto (2019), The Q_2-free process in the hypercube, preprint:

J.R. Johnson, and J. Talbot, Vertex Turan problems in the hypercube, (2010), J Combin. Theory, Ser. A, 117, 454--465

I. Leader and E. Long, Long geodesics in subgraphs of the cube, (2014), Discrete Math. 326, 29--33

N. Morrison, J. Noel and A. Scott, Saturation in the hypercube and bootstrap percolation}, (2017), Combin. Probab. Comput. 26, 78--98

T. Mutze, Proof of the middle levels conjecture, (2016), Proc. LMS, 112, 677--713
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