Category Archives: mathematics

Auckland

Rosemary and I are in Auckland on a seven week research visit, supported by Hood fellowships from the University. Already I am working on several projects, on polytopes, automorphic loops, symmetric designs, optimal neighbour designs, and median graphs. I hope … Continue reading

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Primitive switching classes

Last year I wrote here about switching classes of graphs for which the switching class has a primitive automorphism group. (I repeat the definitions briefly below.) I conjectured that, except for the trivial switching classes of the complete and null … Continue reading

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Automorphism groups of hypergraphs

I am getting old and forgetful, but I don’t think I said anything here about this problem yet. If I did, apologies for the repetition – but there is something new to report! In April, Laci Babai and I finally … Continue reading

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Busy times, 9: Beyond the limit in St Petersburg

Anatoly Vershik is a universal mathematician, with influential work in asymptotic combinatorics, groups and group actions, probability, mathematical physics, and many other areas. This week, I was in St Petersburg for a conference with the wonderful title “Representations, Dynamics, Combinatorics: … Continue reading

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Steiner systems

Following Peter Keevash’s asymptotic existence proof for Steiner systems, does anything remain to be done? I would say yes, it certainly does; here are a few thoughts about the open problems in this area. Existence We are looking for a … Continue reading

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Subsets and partitions

There are several packing and covering problems for subsets of a set, which have been worked over by many people. For example, given t, k and n, how many k-subsets of an n-set can we pack so that no t-subset … Continue reading

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Steiner systems exist

A Steiner system S(t,k,n) is a collection of k-subsets (called “blocks”) of an n-set of “points” with the property that any t-set of points is contained in a unique block. To avoid trivial cases, we assume that t<k<n. Since the … Continue reading

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Definitions, 2

Anyone who has lectured on the construction of the number systems has faced a problem with definition. For the most obvious example, do you define real numbers as Dedekind cuts, or as Cauchy sequences, or more simple-mindedly as infinite decimals, … Continue reading

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Definitions

Time to generalise from the preceding examples. There are several good reasons why a choice of definitions is a good thing. First, as several points in the discussion of graphs suggested, different definitions may be adapted for different kinds of … Continue reading

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Finite geometry and probabilistic combinatorics

In the late 1960s, when I was born as a mathematician, I worked on finite permutation groups, on the edge of finite geometry and the combinatorics of very regular structures. I was dimly aware that there was a completely different … Continue reading

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