# Some old and new problems in combinatorics and geometry

14 Jul

Originally posted on Combinatorics and more:

Paul Erdős in Jerusalem, 1933  1993

I just came back from a great Erdős Centennial conference in wonderful Budapest. I gave a lecture on old and new problems (mainly) in combinatorics and geometry (here are the slides), where I presented twenty problems, here they are:

## Around Borsuk’s Problem

Let $latex f(d)$ be the smallest integer so that every set of diameter one in $latex R^d$ can be covered by $latex f(d)$ sets of smaller diameter. Borsuk conjectured that $latex f(d) \le d+1$.

It is known (Kahn and Kalai, 1993) that : $latex f(d) \ge 1.2^{\sqrt d}$, and also that (Schramm, 1989) $latex f(d) \le (\sqrt{3/2}+o(1))^d$.

Problem 1: Is f(d) exponential in d?

Problem 2: What is the smallest dimension for which Borsuk’s conjecture is false?

## Volume of sets of constant width in high dimensions

Problem 3: Let us denote the volume of the n-ball of radius 1/2 by \$latex…

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