Predavatelj: László A. Székely (University of South Carolina, ZDA)

Katona, Griggs, Lu and many others started a systematic investigation of extremal problems
with excluded posets for the subset lattice, as a far-reaching generalization of Sperner theory.
More precisely: given a fixed partially ordered set, what is the maximum size of a family of
subsets of an n element set, such that the fixed partially ordered set is not realized by inclusion among the members of the family.
The diamond is the partial order of the 4 subsets of a 2 element set. The extremal problem
above is unsolved for the diamond, even in an asymptotic sense, notwithstanding vigorous
work and successive improved bounds. The diamond conjecture is that the threshold is
asymptotically equal to the sum of the sizes of the two largest levels in the subset lattice.
We show that if the diamond conjecture is true, then many different constructions yield it.
We construct them with Markov chains on abelian groups.

This is joint work with Eva Czabarka, Aaron Dutle, and Travis Johnston.