Department of Computer Science | Institute of Theoretical Computer Science | CADMO

Theory of Combinatorial Algorithms

Prof. Emo Welzl and Prof. Bernd Gärtner

Mittagsseminar (in cooperation with M. Ghaffari, A. Steger and B. Sudakov)

Mittagsseminar Talk Information

Date and Time: Tuesday, June 04, 2013, 12:15 pm

Duration: 30 minutes

Location: CAB G51

Speaker: Zuzana Safernová (Charles University)

Helly vs. Betti II

Helly number of a finite family of sets is h if any minimal subfamily with an empty intersection consist of h or fewer sets. (If a family has non-empty intersection then its Helly number is, by convention, 0.) Helly’s theorem then simply states that any finite family of convex sets in Rd has Helly number at most d+1. Such uniform bounds, that is bounds independent of the cardinality of the family, are of particular interest.

We present the following Helly-type result: if the Helly number of a finite family of sets in Rd has huge Helly number then some intersections of the sets must be topologically really complicated (in terms of its Betti numbers).

Precise statement is the following: Suppose that F is a finite family of arbitrary sets in Rd such that the intersection of any subfamily of F has the first d/2 Betti numbers β0,...,βd/2-1 bounded by some number B. Then the Helly number of F is bounded by some number h=h(d,B) that depends only on d and on B.

In the talk we will sketch a proof and say some details, since it is a continuation of a talk given by Uli a month ago.

Joint work with X. Goaoc, P. Paták, M. Tancer and U. Wagner.

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