## Theory of Combinatorial Algorithms

Prof. Emo Welzl and Prof. Bernd Gärtner

# Mittagsseminar (in cooperation with M. Ghaffari, A. Steger and B. Sudakov)

 Mittagsseminar Talk Information

Date and Time: Tuesday, November 14, 2017, 12:15 pm

Duration: 30 minutes

Location: CAB G51

Speaker: Natasha Morrison (University of Oxford)

## Bootstrap Percolation in the Hypercube

The 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 percolates. In this talk I will discuss the proof of a conjecture of Balogh and Bollobas: for fixed r and d going to infinity, the minimum cardinality of a percolating set in the d-dimensional hypercube is $\frac{1+o(1)}{r}\binom{d}{r-1}$. This is joint work with Jonathan Noel.

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