Mittagsseminar (in cooperation with A. Steger, D. Steurer and B. Sudakov)

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

Duration: 30 minutes

Location: OAT S15/S16/S17

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 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 β₀,...,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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