A topological colorful Helly theoremGil KalaiYale and Hebrew University
April 16,

ABSTRACT


Let F be a finite family of convex sets in ndimensional Euclidean space. Helly's theorem asserts that if every n+1 members of the family has a point in common then there is a point in common to all members of the family. Lovasz proved the following extension of Helly theorem: Colorful Helly Theorem. Consider n+1 families of convex sets in R^n and suppose that for every choice of n+1 sets, one from each family there is a point in common to all these sets. Then one of the families is intersecting. This innocent looking extension is quite deeper than Helly's original theorem and the associated Caratheodorytype theorem of Barany has many applications in discrete geometry. Helly himself realized that in his theorem convex sets can be replaced by topological cells if you impose the requirement that all nonempty intersections of these sets are again topological cells. (Since the intersection of convex sets is also convex this requirement is automatically satisfied in the original geometric version.) Helly's topological version of his theorem follows from the later "nerve theorems" of Leray and others. In the lecture I will present a topological version for Lovasz' colorful Helly theorem present a matroidal generalization and discuss related issues. 
Combinatorics Seminar, Mathematics Department, MIT, sara(atsign)math.mit.edu 

