Colin Desmarais

(Department of Mathematics, University of Manitoba)

Finite Geometries and Lower Bounds For $ex(n;K^{(3)}(2,2,t))$

Date Friday, December 5, 2014

For two graphs $F$ and $G$, say that $G$ is $F$-free if G does not contain a copy of $F$ as a subgraph. Given some specified graph $F$ (called a forbidden graph), the $extremal$ $number$ $ex(n; F)$ is the maximum number of edges in any $F$-free graph $G$ on n vertices. The analogous extremal numbers for hypergraphs is of considerable interest. The extremal numbers for a few classes of forbidden graphs is known, while for most, only loose asymptotic bounds are known. This is especially true for classes of forbidden hypergraphs.

In this talk, I will discuss lower bounds for the extremal numbers of a small class of 3-partite 3-uniform hypergraphs. These lower bounds are achieved by constructing very large hypergraphs that exploit properties of finite geometries.