From the archive, 10

Here, for the record, are the other papers in the cache I found this week. Apart from preprints of papers which were published, there are five typescripts and a number of handwritten pages. The unpublished typescripts are:

1. Rosemary A. Bailey and Peter J. Cameron, The eigenvalues of averaged symmetric matrices. [Our first joint paper, rejected and never published. The paper sets the problem of finding the eigenspaces of the n×n matrix obtained by averaging the images of a real symmetric matrix under a transitive group acting on {1,…,n}, and why statisticians want to do this.]
2. Peter J. Cameron, Yet another generalisation of Fisher’s inequality. [Proved in response to a question from one of my first PhD students at Queen Mary, Mark Whelan. An m-part t-design on a set X of points is a collection of functions from X to {1,…,m} such that the inverse images of each point in all functions have the same size, and the number of functions taking prescribed values on t distinct points depend only on the prescribed values and not on the chosen points. For m = 2, this is equivalent to a complementary pair of t-designs. I show that an m-part 2-design has at least (m−1)v functions, where v is the number of points.]
3. Peter J. Cameron, Locally projective spaces and the transitivity of parallelism. Submitted to a journal, possibly Math. Z., in 1976; rejected, then abandoned. [A study of geometries with the property that, if three hyperplanes have the property that two of the three intersections are codimension-2 subspaces, then the third is empty or codimension-2.]
4. Peter J. Cameron, Paired suborbits. [Sims showed that paired subconstituents in a finite transitive permutation group must have a common no-trivial quotient; in my thesis I used his method to prove other results of this sort, and this note extends the ideas further.]
5. Peter J. Cameron, Automorphism groups of parallelisms. [These are parallelisms of the design of all t-subsets of an n-set, where t divides n. Some of this went into my book on parallelisms.]

There are also some handwritten papers:

1. Three things I wrote as an undergraduate: one takes the set of unordered n-tuples over some mathematical structure (ordered set, field, metric space) and explores giving it a related structure; one considers the set of 2×2 real matrices commuting with a given matrix and attempts to mimic the structure of the complex numbers; the third takes the set {1,2,…,mn}, partitions it into m sets of size n, takes the product of the numbers in each set and adds these products, and asks: what is the minimum value?
2. Three different constructions starting from a Hadamard matrix of order n and producing symmetric Hadamard matrices of order n² with constant diagonal and constant row and column sums.
3. A generalisation of the outer automorphism of M₁₂: take a pair of 3-designs, in each of which the complement of a block is a block, and there are correspondences between pairs of points in one design and complementary pairs of blocks in the other.
4. Subsets of projective planes having a constant size intersection with every secant.
5. Yet more generalisations of Fisher’s inequality, some of which got into my BCC papers in 1973 and 1977. One document considers designs in vector spaces; the other, designs in association schemes.
6. Some characterisations of 2-arc transitive graphs (if they have enough quadrangles, they are known: this grew out of my thesis, but unlike there, it did not assume vertex-primitivity.
7. Relations between paired subconstituents of transitive permutation groups. (Sims showed that they must have a common no-trivial quotient; in my thesis I used his method to prove other results of this sort, and this note extends the ideas further.)
8. A paper with Peter Neumann characterising the permutation groups which are 3/2-transitive on the k-subsets of the domain, for some k: the groups S₇ and A₇ are the only examples.

The most interesting to me are the undergraduate pieces, since it might seem as if a lot of the mathematics I did grew from considering the structure of the set of k-subsets of an n-set.