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SUMMARY:Bootstrap Percolation in the Hypercube - Natasha Morrison (Oxford)
DTSTART:20151022T133000Z
DTEND:20151022T143000Z
UID:TALK60767@talks.cam.ac.uk
CONTACT:Andrew Thomason
DESCRIPTION:The \\emph{$r$-neighbour bootstrap process} on a graph $G$ sta
 rts with an initial set of ``infected'' vertices and\, at each step of the
  process\, a healthy vertex becomes infected if it has at least $r$ infect
 ed neighbours (once a vertex becomes infected\, it remains infected foreve
 r). If every vertex of $G$ becomes infected during the process\, then we s
 ay that the initial set \\emph{percolates}. \n\nIn this talk I will discus
 s 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 ke
 y ideas behind the proof exploits a connection between bootstrap percolati
 on and weak saturation. This is joint work with Jonathan Noel.\n
LOCATION:MR12
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