Gil Kalai, The Cascade Conjecture and other Helly-type Problems

[Colloquium, Department of Mathematical Sciences, KAIST]

For a set $X$ of points $x(1)$, $x(2)$, $\ldots$, $x(n)$ in some real vector space $V$ we denote by $T(X,r)$ the set of points in $X$ that belong to the convex hulls of r pairwise disjoint subsets of $X$.
We let $t(X,r)=1+\dim(T(X,r))$.

Radon’s theorem asserts that
If $t(X,1)< |X|$, then $t(X, 2) >0$.

The first open case of the cascade conjecture asserts that
If $t(X,1)+t(X,2) < |X|$, then $t(X,3) >0$.

In the lecture, I will discuss connections with topology and with various problems in graph theory. I will also mention questions regarding dimensions of intersection of convex sets.

Some related material:
1) A lecture (from 1999): An invitation to Tverberg Theorem: https://youtu.be/Wjg1_QwjUos
2) A paper on Helly type problems by Barany and me https://arxiv.org/abs/2108.08804
3) A link to Barany’s book: Combinatorial convexity https://www.amazon.com/Combinatorial-Convexity-University-Lecture-77/dp/1470467097