Black hole coordinates

Perhaps you would explain some of your work. What about your black hole coordinates?

You know, it is probably through my little paper on the black holes that my name is best known to the widest readership, yet I never mentioned 'black holes'! That term didn't exist then. But I must say this was one of the papers which caused me the least effort. Somehow it was in my head, and I think I put the whole thing together in one afternoon. Sometimes you can put an incomparably bigger effort into a paper which affects people much less.

By the way, I tried to persuade the theoretical physicists that what I did there was not much different from the work of an earlier cosmologist, Georges Lemaitre. I could never dig up his footnote where he suggested something similar to what I worked out in this paper, but I did mention his name in my black hole paper.

Actually, I was interested in a much more general problem in geometry: what you should call a singularity of a place. This was at that time a very ill-defined concept, but I gave a method for deciding whether a point is a singularity or not. This was my purpose, and what they later called a 'black hole' was merely a little illustration of the method.

An enduring graph theory problem

And graph theory? Can you explain that fairly simply?

Ye-e-s. In fact, Esther's great problem is a mixed graph theory. Her problem is very simple, as I will explain to you, but the answer is not so simple.

Firstly, this is the most usual way to express graph theory (but it is really much more complex). Say you take a number of points, then you pick out pairs of these points and connect each pair with a line. What you get is a graph.

In normal parlance, to draw a 'graph' means you plot your income or business data, but for the mathematician a graph simply means a visual picture of data. For example, suppose you take 10 points and to each you assign a number, an integer. Say to the first you assign 5 so you put it at a certain point, then to the second you assign another number and you put it at its appropriate point, and so on. Graph theory is the study of that sort of graph.

With that explanation in mind, let us turn to Esther's geometry problem. It is almost a schoolgirl's problem. She noticed that if you take any five points in a plane, then whichever way they are situated in the plane, you can always pick out four which form a convex quadrilateral.

I don't want to give you a high-brow mathematical definition of a convex quadrilateral but on paper I can show you how it differs from a concave one. [Draws] Here are four points; this is a convex quadrilateral. Now I will take another four points but this time they have a triangle inside them. This is concave – simply because if you take the 'envelope' of the whole configuration, this triangle then contains in its interior another point.

What Esther discovered was that if I give you any five points I can always pick out four of them which form a convex quadrilateral. She asked then, innocently, the following problem (which became my not quite solved problem): how many points in the plane do you need so that it should, with certainty, contain say five or six or any specific number of points – k points – which are in a convex configuration?

This turned out really to be a tricky problem. I could prove at that time that if the number of points is big enough, then you can always select a certain number – k – of points which are in a convex configuration. What is troublesome, what is unsolved, is that you don't know how many points in a plane you have to pick out, so that they should always form a basic convex configuration of k points. People have been able to show by experiment what the answer is likely to be, but so far nobody has managed to do it theoretically – minus 2, minus 1, what would it be?

Combinatorics: a new branch of mathematics

What sorts of applications result from solving mathematical problems?

Well, the proof that I gave in my solution to Esther's problem was a bit intricate but it started off a whole branch of combinatorial mathematics. So my great contribution was not so much to solve the problem but to create a new branch of mathematics. It is nowadays called not by my name, unfortunately, but by that of a well-known British mathematician and logician, Ramsey – and even then for quite different reasons. He was interested in the main thing on which my proof rested but I didn't know about his work until later, because mathematicians don't really read logicians' works!

Could you tell me a little bit, then, about combinatorics?

One of the primitive, first problems in combinatorics is shown by this typical high school mathematical problem: if you have an object and you want to pick out k examples of this, what is the number of different ways you can do so? It is very easy to answer that. The magic formula that gives it is l times l-1 times l-2 times l-3 and so on, over k factorial. K factorial is k times k-1 times k-2 and so on. This gives you the number of ways how you can do various things in high school mathematics.

Focus questions

• What do you think is meant by the geometric term 'a singularity in space' and how does this relate to black holes?
• What might be some applications of combinatorics?