From the archive, 4

A photocopy of a sheet of paper in my handwriting. At the top left, my initials are written in the handwriting of Jaap Seidel. The page begins as follows.

Theorem. Let X be a set of points in

the n-cube Qn the Johnson scheme J(k,n) the symmetric group Sn.

Then the average distance between two members of S is not greater than

n/2 2k(n−k)/n n−1.

Equality holds if and only if n is

an orthogonal array of strength 1 a 1-design uniformly transitive.

Then follow three columns giving the proofs of the three propositions (completely elementary, basically Cauchy–Schwarz), followed by some remarks about what conditions a P&Q-polynomial association scheme would have to satisfy for a version of the first two theorems above to hold (basically, the inner product of projections onto the first eigenspace should be a monotone decreasing function of the distance).

There should be more to it, since even such a result would not cover the third case.

And the small mystery: why do I have a photocopy annotated by Jaap? I suspect that I said something like this in conversation at a conference, Jaap was interested, so I wrote out a proof and gave it to him, and he decided firmly that I should have a copy myself. When could this have been? My guess is the Montréal conference on algebraic, extremal and metric combinatorics in the late 1980s, but I am not certain.

About Peter Cameron

I count all the things that need to be counted.
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