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CATEGORIES:Combinatorics Seminar
SUMMARY:Families with few k-chains - Adam Zsolt Wagner (Un
iversity of Illinois Urbana-Champaign)
DTSTART;TZID=Europe/London:20170223T143000
DTEND;TZID=Europe/London:20170223T153000
UID:TALK70607AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/70607
DESCRIPTION:A central theorem in combinatorics is Sperner's Th
eorem\, which determines the maximum size of a fam
ily in the Boolean lattice that does not contain a
2-chain. Erdos later extended this result and det
ermined the largest family not containing a k-chai
n. Erdos and Katona and later Kleitman asked how m
any such chains must appear in families whose size
is larger than the corresponding extremal result.
\n\nThis question was resolved for 2-chains by Kle
itman in 1966\, who showed that amongst families o
f size M in the Boolean lattice\, the number of 2-
chains is minimized by a family whose sets are tak
en as close to the middle layer as possible. He al
so conjectured that the same conclusion\nshould ho
ld for all k\, not just 2. The best result on this
question is due to Das\, Gan and Sudakov who show
ed roughly that Kleitman's conjecture holds for fa
milies whose size is at most the size of the k+1 m
iddle 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 hol
ds for families of size at most (1-epsilon)2^n\, t
hereby establishing Kleitman's conjecture\nasympto
tically. Our proof is based on ideas of Kleitman a
nd Das\, Gan and Sudakov.\n\nJoint work with Jozsi
Balogh.\n
LOCATION:MR12
CONTACT:Andrew Thomason
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