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# Category Archives: open problems

## 9,21,27,45,81,153,…

This is the sequence of degrees of primitive groups which don’t synchronize a map of rank 3, equivalently graphs with clique number and chromatic number 3 having primitive automorphism groups. You could argue that the sequence should start with 3, … 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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## 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

## A small problem

Infinite products are an attractive part of real analysis which has fallen out of many syllabuses. I am concerned here only with infinite products in which the factors are between 0 and 1. The partial products are positive and decreasing, … Continue reading

## 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
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## 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