Abstract:

Over the last three decades I am having a sort of mathematical dialogue with Jürgen Eckhoff and I will devote the lecture to mention some of its highlights.

1) In 1980 both Eckhoff and I proved by different methods an "upper bound conjecture" of Katchalski and Perles.

2) In 1983 I proved a conjecture of Eckhoff giving a complete description of f-vectors of nerves of families of convex sets in Euclidean spaces.

3) In 1988 Eckhoff proved a conjecture of mine regarding families of standard boxes in Euclidean spaces.

4) Around 1990 we both formulated separately a conjecture on face numbers of flag complexes (number of complete subgraphs of various sizes of a graph) which was proved in 2005 by Andy Frohmader. Frohmader's theorem asserts that the f-vector of a clique complex (the complex of complete subgraphs of a graph) (aka "flag complex") of dimension d is the f-vector of some d-colored simplicial complex. This is a far reaching extension of Turan's theorem.

5) In 2006 Meshulam and me proved a result about homological dimensions of projections of simplicial complexes which implies a topological version of a Helly type theorem by Amenta. In the same year Eckhoff and Klaus-Peter Nischke proved a far reaching combinatorial extension of Amenta's theorem first offered by Morris. Their result also implies the same topological extension of Amenta's theorem.

6) Eckhoff and I proposed in the mid 70s (different) far reaching extensions of Tverberg's theorem. Both are still open.

There are other points: The work of Björner and me on face numbers and Betti numbers of cell complexes is closely related to earlier works by Eckhoff and Wegner on the Kruskal-Katona theorem; The work by Alon and me on (p,q) theorems for hyperplane transversals is also strongly related to earlier works by Eckhoff.