One of the hardest things for a supervisor of a new doctoral student is to choose a research topic. It should be one which is not trivially easy or impossibly difficult. But of course it is impossible to say in advance whether a problem will fall into one of these categories. It is especially difficult with your first student, when you have no experience to fall back on.
So I was extremely lucky that my first student, Robert Peile, was sufficiently independent-minded that he chose his own topic. I thought it was going to be tough, and tried to persuade him onto something easier, but he stood firm.
Here is a brief account. A few years earlier, Graham Higman had made the following observation. Let S be a collection of 3-element subsets of a finite set V. Then the following conditions are equivalent:
- any 4-element subset of Vcontains an even number of members of S;
- there is a graph on the vertex set V with the property that the sets in S are exactly the 3-element sets which contain an odd number of edges of the graph.
These are the objects to which Higman gave the name two-graphs. The “two” indicated that (V,S) is a 2-dimensional simplicial complex.
This observation can be re-phrased as follows. For positive integers k and l with k < l, let M(k,l) be the incidence matrix of k-sets and l-sets of V; that is, rows are indexed by k-sets, columns by l-sets, with (K,L) entry 1 if K is a subset of L and 0 otherwise. Then, if we work over the 2-element field, then the kernel of M(3,4) is equal to the image of M(2,3).
It is not difficult to show that the same holds if (2,3,4) are replaced by three consecutive integers. But Robert’s plan was to take any pair (k,l) and describe the kernel of M(k,l) in terms of the images of M(k,k) for an appropriate collection of values of i.
Despite my cynicism, he succeeded in his quest, and earned his DPhil degree. Then he took a position with Racal and later had some connection with the University of Southern California. He co-authored a couple of things with Solomon Golomb, including a book on information and coding with the beautiful subtitle “The Adventures of Secret Agent 00111”. He did publish the results in his thesis in a paper entitled “Inclusion transformations: (n,m)-graphs and their classification” in Discrete Mathematics in 1991. (As a footnote, Fan Chung and Ron Graham published a similar paper called “Cohomological aspects of hypergraphs” in the Transactions of the American Mathematical Society in 1992.)
He died, much too soon, in 1995.
Robert was haemophiliac, and needed regular blood transfusions. But he let this stand as little as possible in the way of his achieving what he set out to do. But when blood contaminated with AIDS and hepatitis viruses was imported from the USA and given to patients, Robert was infected (I think with AIDS) and died.
Now, belatedly, a report on the contaminated blood affair has just been released in Britain. It has strong things to say about the actions of politicians and health workers (with a few honorable exceptions). This brought Robert back to my mind. But even prompt action then would have been too late for him.
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