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SUMMARY:Independent sets in hypergraphs - Wojciech Samotij (University of 
 Cambridge)
DTSTART:20111027T133000Z
DTEND:20111027T143000Z
UID:TALK34096@talks.cam.ac.uk
CONTACT:Andrew Thomason
DESCRIPTION:We say that a hypergraph is _stable_ if each sufficiently larg
 e subset of its vertices either spans many hyperedges or is very structure
 d. Hypergraphs that arise naturally in many classical settings possess the
  above property. For example\, the famous stability theorem of Erdős and 
 Simonovits and the triangle removal lemma of Ruzsa and Szemerédi imply th
 at the hypergraph on the vertex set $E(K_n)$ whose hyperedges are the edge
  sets of all triangles in $K_n$ is stable. In the talk\, we will present t
 he following general theorem: If $(H_n)_n$ is a sequence of stable hypergr
 aphs satisfying certain technical conditions\, then a typical (i.e.\, unif
 orm random) $m$-element independent set of $H_n$ is very structured\, prov
 ided that $m$ is sufficiently large. The above abstract theorem has many i
 nteresting corollaries\, some of which we will discuss. Among other things
 \, it implies sharp bounds on the number of sum-free\nsets in a large clas
 s of finite Abelian groups and gives an alternate proof of Szemerédi's th
 eorem on arithmetic progressions in random subsets of integers.\n\nJoint w
 ork with Noga Alon\, József Balogh\, and Robert Morris.\n
LOCATION:MR12
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