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I am a quartertime Professor in the School of Mathematics and Statistics at the University of St Andrews, and an Emeritus Professor of Mathematics at Queen Mary, University of London. In addition, I am an associate researcher at CEMAT, University of Lisbon, Portugal.
I am a Fellow of the Royal Society of Edinburgh.
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A base for a permutation group is a sequence (a_{1},…,a_{r}) of points in the permutation domain whose pointwise stabiliser is the identity. A base is irredundant if no point in the base is fixed by the stabiliser of its predecessors; it is minimal if no point in the base is fixed by the stabiliser of all the others.
A couple of things are clear:
In 1995, Dima FonDerFlaass and I showed that the three conditions on irredundant bases, namely
Recently several people (Murray Whyte, Scott Harper, Pablo Spiga, maybe others) asked whether the set of sizes of irredundant bases form an interval. This is true. However, the analogue for minimal bases is false. For any integer d > 1, there is a group whose minimal bases have sizes 1 and d only. However, it is not yet known whether they do form an interval if the permutation group is required to be transitive.
A further question is: What happens for greedy bases? One of these is found by choosing the next base point in an orbit of maximum size for the stabiliser of its predecessors. (There is some variability because there might be a number of orbits of the same size.)
Old research snapshots are kept here.
I am Honorary EditorInChief of the Australasian Journal of Combinatorics, an international openaccess journal published by the Combinatorial Mathematics Society of Australasia. 
School of Mathematics and Statistics
University of St Andrews North Haugh St Andrews, Fife KY16 9SS SCOTLAND 
Tel.: +44 (0)1334 463769 Fax: +44 (0)1334 46 3748 Email: pjc20(at)starthurs(dot)ac(dot)uk [oops – wrong saint!] 
Page revised 13 April 2023 
A (total) ordering of a finite vector space is called compatible if every orderpreserving bijection between subspaces of the same dimension is linear.
It is straightforward to show that every finite vector space has a compatible ordering: for example, choose any ordering of the finite field F; then the lexicographic ordering of F^{n} is compatible, for any n.
Problem: How many compatible orderings does the ndimensional vector space over the field of order q have? (Formulae are known for the cases n = 1 and n = 2.)
Old problems are kept here.