Bootstrap Percolation in the Hypercube

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• Natasha Morrison (Oxford)
• Thursday 22 October 2015, 14:30-15:30
• MR12.

If you have a question about this talk, please contact Andrew Thomason.

The \emph{$r$-neighbour bootstrap process} on a graph $G$ starts with an initial set of ``infected’’ vertices and, at each step of the process, a healthy vertex becomes infected if it has at least $r$ infected neighbours (once a vertex becomes infected, it remains infected forever). If every vertex of $G$ becomes infected during the process, then we say that the initial set \emph{percolates}.

In this talk I will discuss the proof of a conjecture of Balogh and Bollob\’{a}s: for fixed $r$ and $d\to\infty$, the minimum cardinality of a percolating set in the $d$-dimensional hypercube is $\frac{1+o(1)}{r}\binom{d}{r-1}$. One of the key ideas behind the proof exploits a connection between bootstrap percolation and weak saturation. This is joint work with Jonathan Noel.

This talk is part of the Combinatorics Seminar series.

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