Discrete Applied Mathematics Seminar by Michael Tait: Subgraphs of Pseudo-Random Graphs

Speaker: Michael Tait, Villanova University

Title: Subgraphs of Pseudo-Random Graphs
Abstract: Fix a graph \(F\). If I have a large pseudo-random graph can I always find \(F\) as a subgraph? Can I count the number of copies? What about if I only have a sufficiently large subset of vertices of my pseudo-random graph?
We discuss how to use spectral graph theory to give a general framework for questions like these. Some of the applications are to questions in discrete geometry and VC dimension. One representative question that we are motivated by is the following:

Let \(E\) be a set in \(\mathbb{F}_q^d\) and \(\alpha, \beta\in \mathbb{F}_q^*\). How large does \(E\) need to be to guarantee that there are four points \(w, x, y, z\in E\) such that they form a rectangle of side lengths \(\alpha\) and \(\beta\), i.e.
\begin{equation}(w-x)\cdot (x-y)=0, ~(x-y)\cdot (y-z)=0, ~(y-z)\cdot(z-w)=0, ~(z-w)(w-x)=0,\end{equation}
and \begin{equation}||w-x||=||y-z||=\alpha, ~||x-y||=||z-w||=\beta.\end{equation}

Our general framework makes progress on this and other similar questions as a corollary. This is joint work with Thang Pham, Steven Senger, and Vu Thi Huong Thu (and if time permits also with Alex Iosevich and Thu Huyen Nguyen).

Discrete Applied Math Seminar

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