Fejes Toth Lecture

One old conjecture of mine was proved by Elekes using a brand new method of his involving polynomials. This method has been adopted by others and is being applied to, among other things, two other problems of Erdos and myself.

A student Justin Smith and I have extended a result by P.D.T.A. Elliott on the number of circles determined by n points to find the number of spheres determined by n points, and our solution reduces the problem to the Orchard Problem. Any improvement in our bound would imply an improvement in the Orchard Problem, and vice versa. The solution to both problems is therefore known within a narrow range, but not known exactly.

I made a conjecture 25 years ago about the number of hyperplanes determined by n points in E^d if certain degeneracies are avoided. I proved this in a weakened form for d = 3 when the Szemeredi-Trotter result became available as a tool. Here is the result:

If L is the number of lines determined by n points and p is the number of planes, then p > c L for some c > 0, provided the points are not all contained on two skew lines or on one plane.

Recently a student Ben Lund and I disproved my conjecture when d = 4. This led us to modify the conjecture by making more degenerate configurations illegal. In a lengthy research program aimed at proving this modified conjecture, we are developing tools. I turns out that Szemeredi-Trotter in higher dimensions is not the way to go, but rather we think the right approach is to extend the Beck-Erdos Theorem to higher dimensions and to prove a bichromatic version of a theorem of Agarwal and Aronov. We have succeeded in proving the latter and are continuing to work on the conjecture.