Discrete Math Seminar: Eva Czabarka

Sperner's theorem from 1928 states that the greatest number subsets of an $n$-element set such that no subset contains another (in other words the largest chain is length $1$), is $\binom{n}{\lfloor n/2\rfloor}$. This result has many generalizations since: $L$-Sperner families are families where the largest chain is of length at most $L$, $M$-part families are families where there is no chain of length $2$ where the increase of a chain is contained in a fixed $M$-partition of the underlying set, etc. Mixed orthogonal arrays are designs introduced by statisticians for designing experiments, so that factors potentially influential to the outcome occur simultaneously in a regular manner. We show that these distant topics have a strong connection (in particular mixed ortogonal arrays and homogeneous $M$-part $(L_1,...,L_M)$-Sperner families correspond to each other), and provide constructions for mixed orthogonal arrays.

Joint work with H Aydinian and L.A. Szekely