## The Convex Hull Method I.

- Oct. 17, 2012
- 3:30 p.m.
- LeConte 312

## Abstract

The entries of the profile matrix of a set family give the number of elements of a given size in the set family. This concept can be extended to $M$-part set families giving rise to $M$ dimensional arrays. These arrays then can be viewed as vectors in an appropriate higher dimensional space. The vertices of the convex hull of profile matrices of certain families of sets were described by P.L. Erdős, P. Frankl, and G.O.H. Katona in 1985, facilitating the optimization of linear functions of the entries of profile matrices of members of the family in question. In 1986 P.L. Erdős and G.O.H. Katona adapted the method for $M$-part Sperner families. In the upcoming lectures I will introduce the method, sketch the proof and show our recent extensions to $M$-part $k$-dimensional Sperner multi-families satisfying certain extra conditions. Our results have new consequences even to $M$-part Sperner families that were not covered by the Erdős-Katona result. Joint work with H. K. Aydinian andL.A. Székely