Discrete Mathematics 2010-2022

\section*{Programm:}
\begin{description}

\item{10:00-11:00} Peter Kritzer (JKU Linz)

Quasi-Monte Carlo using points with non-negative local discrepancy}

\item{11:00-12:00} Christoph Aistleitner (TU Graz)

TBA}

\item{12:00-14:00} Gemeinsames Mittagessen

\item{14:00-15:00} Martin Ehler (Universität Wien)

TBA}

\item{15:00-16:00} Josef Dick (University of New South Wales, Sydney)

TBA}

\end{description}

Bootstrap percolation} is a spreading process on graphs where starting with an initial set $A$ of infected vertices, other vertices become infected once a certain threshold $r$ of their neighbors are infected. We say that the set $A$ percolates if eventually all vertices of the graph are infected. When $r = d(v)/2$, where $d(v)$ is the degree of a vertex, that is, when a vertex becomes infected once half of its neighbors are already infected, we call the process majority bootstrap percolation}.

We analyse this process when the initial set $A$ is chosen randomly, with each vertex being assigned to $A_p$ with a fixed probability $p$ independently, which in some way represents the ‘typical’ behaviour for sets of density $p$. Balogh, Bollob\'{a}s and Morris considered this process on the hypercube, a well-studied geometric graph, and demonstrated a sharp threshold for the property that the set $A_p$ percolates. We determine a similar sharp threshold for a broader class of product graphs}, those arising as the cartesian product of many graphs of fixed order.

Joint work with Mauricio Collares, Joshua Erde and Mihyun Kang.