## Supersaturation in the Boolean Lattice

- Nov. 7, 2012
- 3:30 p.m.
- LeConte 312

## Abstract

Turan's theorem begs the question: given fixed integers $n$ and $m$ and a fixed graph $H$, over all graphs with $n$ vertices and $m$ edges, which has the minimum number of subgraphs isomorphic to $H$? There is an analogous question for the Boolean lattice: given fixed integers $n$ and $m$ and fixed poset $P$, for all families $F$ contained in $B(n)$ where $|F|=m$, which family has the minimum number of subposets of $F$ isomorphic to $P$? In 1966, Kleitman answered this question for $P$ being a single edge. I will be outlining this proof as well as discussing possible further advances in answering this question.