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# Category Archives: open problems

## 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

## 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

Posted in mathematics, open problems
Tagged primitivity, sections, semigroups, transversals
1 Comment

## A Cayley graph challenge

Greg Cherlin showed that Henson’s graphs are Cayley graphs, so perhaps it is time to look again at the question: Is Covington’s graph a Cayley graph? Here, to start things off, is a simple fact: Covington’s graph G is not … Continue reading

## The sound of problems falling

This month brought news that two problems I posed have been solved. A conjecture of mine was proved by Martin Bridson and Henry Wilton, and another question (which I didn’t feel brave enough to connjecture) has been answered by Greg … Continue reading

## Carries, shuffling, and cocycles

Last week we were treated to a lovely lecture by Persi Diaconis. As he so often does, he started with an elementary question: how many carries do you expect if you add n numbers in base b? From there he … Continue reading

Posted in exposition, open problems
Tagged arithmetic, minimal cocycles, riffle shuffles, Seidel, switching
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## A Shrikhande challenge

I discussed here the problem of covering the m-fold complete graph on n vertices with copies of a given graph G. The smallest strongly regular graph for which I don’t know the answer is the Shrikhande graph. I can copy … Continue reading

## Symmetry versus regularity

In my report on CAMconf, I didn’t mention Laci Babai’s talk, whose title was the same as that of this post. This was a talk that needed some thinking about. I want to describe the situation briefly, and then pose … Continue reading

Posted in events, exposition, open problems
Tagged primitive group, Seidel switching, Steiner system, strongly regular graph, switching class
4 Comments

## Partitions into Petersens

There is a lovely algebraic argument to prove that the complete graph on ten vertices (which has 45 edges) cannot be partitioned into three copies of the Petersen graph (which has 15 edges). Sebastian Cioaba asked me: for which m … Continue reading

Posted in exposition, open problems
Tagged doubly transitive groups, eigenvalues, Ramsey's theorem
9 Comments

## A problem and a bet

In the young but flourishing subject of permutation patterns, it is usual to regard a permutation of {1,…,n} as simply a sequence containing each of these numbers just once. So 31542 is a permutation of {1,…,5}. This is in agreement … Continue reading

## Endomorphism monoids of graphs

A monoid is, for me, a set of mappings on a finite domain which is closed under composition and contains the identity mapping. The composition is, of course, associative. Thus, it is “a group without the inverses”. A homomorphism from … Continue reading

Posted in exposition, open problems
Tagged endomorphism monoid, grid graph, Latin square, semigroup
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