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# Category Archives: open problems

## Association schemes for diagonal groups

Sean Eberhard commented on my posts on diagonal groups (see here and here). He is correct; there is an association scheme preserved by the full diagonal group with n factors in the socle; it is non-trivial if n > 2. The details … Continue reading

Posted in open problems
Tagged association scheme, diagonal group, Latin hypercube, Latin square
1 Comment

## Fair games and Artin’s conjecture

A few years ago I described Persi Diaconis’ response to G. H. Hardy’s claim that there is a real dividing line between real and recreational mathematics. (See the report here.) This led from Persi’s first experiments in card shuffling to Artin’s conjecture … Continue reading

## Kourovka Notebook, 19th edition

The latest edition (the 19th) of the Kourovka Notebook has just been released. It now has its own website, https://kourovka-notebook.org/. The Kourovka Notebook has been going for more than 50 years, longer than my life as a mathematician. It is … Continue reading

Posted in doing mathematics, history, open problems
Tagged identical relations, Kourovka Notebook, open problems
1 Comment

## Polynomially bounded orbit counts

The best news I had yesterday was an email from Justine Falque with a link to a paper that she and Nicolas Thiéry have just put on the arXiv. The 12-page document is only the “short version”, and a longer … Continue reading

## A problem

Since I have been saying rather a lot about association schemes and coherent configurations lately, I thought I would mention an open problem. This is probably one for the experts, and I guess it has been ignored because of the … Continue reading

Posted in open problems
Tagged association scheme, coherent configuration, permutation group
8 Comments

## Outer automorphisms

I have just put on the arXiv a paper I wrote with Sam Tarzi ten years ago. I want to say here something about the context, the contents of the paper, and the reason for posting it now. Outer automorphisms … Continue reading

## A small fact about the Petersen graph

The Petersen graph has 10 vertices and 15 edges, and the complete graph on 10 vertices has 45 edges. However, Allen Schwenk and (independently) O. P. Lossers (Jack van Lint’s problem-solving seminar in Eindhoven) showed that you can’t partition the … Continue reading

## Algebraic properties of chromatic polynomials, 2

My paper with Kerri Morgan on algebraic properties of chromatic roots (described here) has just appeared in the Electronic Journal of Combinatorics: you can find it here. I won’t say any more about it, except to pose a challenge which … Continue reading

Posted in open problems, Uncategorized
Tagged algebraic integer, chromatic polynomial, Galois group
2 Comments

## Automorphism groups of transformation semigroups

I have been in the transformation semigroups game for nearly ten years now, but I still feel that I am finding my feet. Here is apparently a huge difference between permutation groups and transformation semigroups, one which is still not … Continue reading

Posted in doing mathematics, open problems, Uncategorized
Tagged automorphism, primitive groups, synchronization
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## Problems

I have begun the long job of updating my collection of open problems. I would appreciate any help! My St Andrews problems are here, and are in pretty good shape (there are only 14 of them so this wasn’t a … Continue reading