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

Prof. Emo Welzl and Prof. Bernd Gärtner

Mittagsseminar Talk Information |

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

**Duration**: 30 minutes

**Location**: OAT S15/S16/S17

**Speaker**: Zuzana Safernová (Charles University)

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 **R**^{d} 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 **R**^{d} 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 **R**^{d} such that the intersection of any subfamily of *F* has the first *d*/2 Betti numbers β_{0},...,β_{d/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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