## Turán Problems on Non-uniform Hypergraphs

- Sept. 19, 2012
- 3:30 p.m.
- LeConte 312

## Abstract

Motivated by extremal poset problems, we will study the Turán
problems on non-uniform hypergraphs. A (non-uniform) hypergraph $H$ is a
pair $(V,E)$ with the vertex set $V$ and the edge set $E\subseteq
2^{V}$. Here we have no restriction on the cardinalities of edges.
The set $R(G):=\{|F|\colon F\in E\}$ is called the set of its edge
types. For a non-uniform hypergraph $G$ on $n$ vertices, we define the
Lubell function of $G$ as
$h(G)=\sum _ {F\in E(G)}\frac{1}{\binom{n}{|F|}}.$

For a given hypergraph $H$, we study the extremal hypergraphs
with maximum Lubell values $h(G)$ among all $H$-free hypergraphs $G$ of
the same edge types of $H$. (This a preliminary report, joint work with
Travis Johnston.)