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

Prof. Emo Welzl and Prof. Bernd Gärtner

Mittagsseminar Talk Information |

**Date and Time**: Tuesday, March 27, 2018, 12:15 pm

**Duration**: 30 minutes

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

**Speaker**: Wojciech Samotij (Tel Aviv University)

A well-known theorem of Sperner describes the largest collections of subsets of an *n*-element set none of which contains another set from the collection. Generalising this result, Erdős characterised the largest families of subsets that do not contain a chain of sets of an arbitrary length *k*. The extremal families contain all subsets whose cardinalities belong to an interval of length *k*–1 centred around *n*/2. In a far-reaching extension of Sperner's theorem, Kleitman determined the smallest number of chains of length two that have to appear in a collection of a given number a of subsets. For every *a*, this minimum is achieved by the collection comprising a sets whose cardinalities are as close to *n*/2+1/4 as possible. Kleitman conjectured that the same is true about chains of an arbitrary length *k*, for all *a* and *n*. We will sketch a proof of this conjecture.

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