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BEGIN:VEVENT
UID:06a72a1c9e5a40ad6e2c55dec5f1deb7
CATEGORIES:Special Colloquium
CREATED:20211129T161812
SUMMARY:From Janson's inequality to hypergraph containers
LOCATION:Zoom
DESCRIPTION:Abstract:Start with a complete graph with n vertices, and choose m edges un
 iformly at random. This basic model of random graphs is known as the Erd?s–
 Rényi G(n,m) model. One of the fundamental questions is how does the probab
 ility that G(n,m) does not contain some chosen graph H change as m goes fro
 m 1 to ex(H, n), the largest number of edges in an H-free graph. A standard
  approach to this question is via Janson's inequality. Surprisingly, and in
  contrast to the binomial random graph model G(n,p), this approach does not
  always correctly determine the order of the logarithm of the probability w
 hen H is a bipartite graph. In particular, a recent result of Balogh, Morri
 s, and Samotij shows that in this case the probability eventually becomes s
 uper-exponentially small in m, whereas Janson's inequality reaches a platea
 u at an exponential bound.\nIn the first part of the talk we discuss a new 
 proof of this result and its connection to the celebrated K?R conjecture. I
 n the second part we discuss applications of these results in Ramsey theory
  and the motivation for revisiting the problem. Finally, generalising the p
 resented ideas, we give a strengthening of Janson's inequality in the spiri
 t of the so-called hypergraph containers, one of the most influential resul
 ts in probabilistic combinatorics in the last decade.\n
X-ALT-DESC;FMTTYPE=text/html:<p>Abstract:Start with a complete graph with n vertices, and choose m edges
  uniformly at random. This basic model of random graphs is known as the Erd
 ?s–Rényi G(n,m) model. One of the fundamental questions is how does the pro
 bability that G(n,m) does not contain some chosen graph H change as m goes 
 from 1 to ex(H, n), the largest number of edges in an H-free graph. A stand
 ard approach to this question is via Janson's inequality. Surprisingly, and
  in contrast to the binomial random graph model G(n,p), this approach does 
 not always correctly determine the order of the logarithm of the probabilit
 y when H is a bipartite graph. In particular, a recent result of Balogh, Mo
 rris, and Samotij shows that in this case the probability eventually become
 s super-exponentially small in m, whereas Janson's inequality reaches a pla
 teau at an exponential bound.</p><p>In the first part of the talk we discus
 s a new proof of this result and its connection to the celebrated K?R conje
 cture. In the second part we discuss applications of these results in Ramse
 y theory and the motivation for revisiting the problem. Finally, generalisi
 ng the presented ideas, we give a strengthening of Janson's inequality in t
 he spirit of the so-called hypergraph containers, one of the most influenti
 al results in probabilistic combinatorics in the last decade.</p>
CONTACT:Rajko Nenadov, Google Zurich
X-EXTRAINFO:This talk is for the local Rutgers Math Community only. Zoom links will be 
 sent by the Department Chair via email
DTSTAMP:20260901T042708
DTSTART;TZID=America/New_York:20211203T113000
DTEND;TZID=America/New_York:20211203T123000
SEQUENCE:0
TRANSP:OPAQUE
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