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 The unpublished typescripts are:
- 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.]
- 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.]
- 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.]
- 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.]
- 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:
- 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?
- 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.
- 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.
- Subsets of projective planes having a constant size intersection with every secant.
- 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.
- 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.
- 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.)
- 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.