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SUMMARY:Families with few k-chains - Adam Zsolt Wagner (University of Illi
 nois Urbana-Champaign)
DTSTART:20170223T143000Z
DTEND:20170223T153000Z
UID:TALK70607@talks.cam.ac.uk
CONTACT:Andrew Thomason
DESCRIPTION:A central theorem in combinatorics is Sperner's Theorem\, whic
 h determines the maximum size of a family in the Boolean lattice that does
  not contain a 2-chain. Erdos later extended this result and determined th
 e largest family not containing a k-chain. Erdos and Katona and later Klei
 tman asked how many such chains must appear in families whose size is larg
 er than the corresponding extremal result.\n\nThis question was resolved f
 or 2-chains by Kleitman in 1966\, who showed that amongst families of size
  M in the Boolean lattice\, the number of 2-chains is minimized by a famil
 y whose sets are taken as close to the middle layer as possible. He also c
 onjectured that the same conclusion\nshould hold for all k\, not just 2. T
 he best result on this question is due to Das\, Gan and Sudakov who showed
  roughly that Kleitman's conjecture holds for families whose size is at mo
 st the size of the k+1 middle layers of the Boolean lattice. Our main resu
 lt is that for every fixed k and epsilon\, if n is sufficiently large then
  Kleitman's conjecture holds for families of size at most (1-epsilon)2^n\,
  thereby establishing Kleitman's conjecture\nasymptotically. Our proof is 
 based on ideas of Kleitman and Das\, Gan and Sudakov.\n\nJoint work with J
 ozsi Balogh.\n
LOCATION:MR12
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