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# Category Archives: exposition

## A cliff

The “combinatorial explosion” is a well-known phenomenon. I recently came across a very dramatic example of it. I was trying to compute the function F(n,k), defined to be the maximum of |S|×|P|, over all sets S of k-subsets and all … Continue reading

Posted in exposition
Tagged combinatorial explosion, computers, GAP, GRAPE, Leonard Soicher, partitions, subsets, transversals
4 Comments

## Groups, lattices and bases

About ten years ago I wrote a six-page paper, which I didn’t succeed in getting any journal editor to publish. I will say a bit about its contents below, but you can read it now: I have posted it on … Continue reading

## Easy to state, hard to solve?

I described here how Pablo Spiga and I showed that all but finitely many nontrivial switching classes of graphs with primitive automorphism group contain a graph with trivial automorphism group, and found the six exceptions. (The trivial switching classes are … Continue reading

Posted in exposition, open problems
Tagged graphs, homomorphisms, primitive groups, rigid graphs, switching classes, tournaments
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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

Posted in exposition, mathematics
Tagged Akos Seress, Don Taylor, Pablo Spiga, primitive group, switching class, two-graph
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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

Posted in exposition, mathematics
Tagged Akos Seress, hypergraphs, Laci Babai, Pablo Spiga, primitive groups
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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

Posted in events, exposition, mathematics
Tagged Anatoly Vershik, asymptotic combinatorics, Ramsey theory, St Petersburg, string theory
6 Comments

## Categorification, step 1

Today at the St Petersburg meeting, Igor Frenkel talked about categorification. He explained that there are five levels (maybe more!) and one has to take certain steps between them; he illustrated with an example, where level 0 was Jacobi’s Triple … Continue reading

## Terminology: association scheme or coherent configuration?

At the Villanova conference, many of the talks were about association schemes or coherent configurations, or indeed generalisations of these. A certain tension between different uses of these terms was evident. I’d like to set down my own views here. … Continue reading

Posted in exposition, history
Tagged association scheme, Bose, coherent configuration, Higman, Leman, Mesner, Nair, Schur ring, Shimamoto, Terwilliger, Weisfeiler
2 Comments

## Futoshiki squares

Futoshiki is a puzzle in the spirit of sudoku, involving constructing a Latin square from some partial information, which can be found in some newspapers now, including the Saturday Guardian. Probably someone has looked at the mathematics of futoshiki, but … Continue reading

## Busy times, 2

Some progress since the last report: the marking is nearly finished! This is just as well, since they have spent the morning drilling in the courtyard and I can’t stand a lot more of it. At first it was rather … Continue reading