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I am a halftime 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.
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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 20 February 2015 
Given n and k with k < n, what is the smallest number d with the following property: given a set S of permutations of an nset X generating a group G, and two ksubsets A and B of X in the same Gorbit, there is a product of at most d elements of S mapping A to B.
For k = 1, it is clear that n−1 suffices, and it is easy to construct examples showing this to be best possible.
I'm really interested in k = 2. Here n(n−1)/2−1 suffices, but I conjecture that the true value is much smaller (maybe a factor of 2 smaller?)
Old problems are kept here.