Mittagsseminar (in cooperation with J. Lengler, A. Steger, and D. Steurer)

Date and Time: Tuesday, October 08, 2019, 12:15 pm

Duration: 30 minutes

Location: OAT S15/S16/S17

Speaker: István Tomon

Partitioning the Boolean lattice into chains of uniform size

The Boolean lattice 2^[n] is the family of subsets of [n]={1,...,n} ordered by inclusion, and a chain in 2^[n] is a family {A_1,...,A_k} such that A_j contains A_i for i < j. By the well known theorem of Sperner (1928), the minimum number of chains 2^[n] can be partitioned into is equal to the number of subsets of [n] of size n/2. However, not much is known about the chain partitions achieving this minimum. Addressing a conjecture of Furedi (1985), we prove that there exists such a chain partition where the sizes of any two chains are within a constant factor, in particular, each chain has size roughly n^1/2. I will talk about how such uniform chain partitions can be used to solve certain extremal and partitioning problems in the Boolean lattice.

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