IMI Interdisciplinary Mathematics InstituteCollege of Arts and Sciences

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.)

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