« Helly-type Problems: Topology, the Cascade Conjecture, and Graph Coloring
Helly-type Problems: Topology, the Cascade Conjecture, and Graph Coloring
November 01, 2021, 2:00 PM - 3:00 PM
Location:
Hill Center-Room 705
Gil Kalai, Hebrew University of Jerusalem
*Introductory 16-minute video (not required for the talk itself; if it does not load, try a different browser) https://idc-il.zoom.us/rec/share/DwDhgNJ5JOJt24ZgFOLUsD_ja5bIsoteWbjB3ujcb8pkIUjs--R7f43apwEBDyE.5n8Sff_mq6we4jsq
Helly-type theorems and problems form a nice area of discrete geometry. In the lecture I will start with the notable theorems of Radon and Tverberg and mention the following conjectural
extension.
For a set X of points x(1), x(2),...,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 graph coloring.