Category Archives: open problems

unsolved mathematical problems

Projective and polar spaces

I have produced a new edition of my lecture notes on Projective and Polar Spaces and put them with my lecture note collection. I did this because it seems that people still find some use for these notes. According to … Continue reading

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Chains of semigroups

I have written here about the lovely formula for the length of the longest chain of subgroups in the symmetric group Sn: take n, increase it by 50% (rounding up if necessary), subtract the number of ones in the base … Continue reading

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A niggling problem

Preparing my talk for the Research Day, I was reminded of a problem I can’t solve, that has niggled me for many years. Maybe someone else can solve it, or maybe I will be encouraged to do it myself. Here … Continue reading

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Bijective proofs

A fourth proof Last month I described three proofs of the formula for the number of ways to choose k objects from a set of n, if repetition is allowed and order is not significant; it is the same as … Continue reading

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Three versus four

The title is homage to the Gödel’s Last Letter and P=NP blog, which a week ago had a post entitled Two versus three. At the problem session at the Banff meeting last night, Dugald Macpherson and Andras Pongracz posed various … Continue reading

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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

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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

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