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CATEGORIES:Combinatorics Seminar
SUMMARY:Minimum saturated families of sets - Matija Bucic
(ETH Zurich)
DTSTART;TZID=Europe/London:20180503T160000
DTEND;TZID=Europe/London:20180503T170000
UID:TALK103615AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/103615
DESCRIPTION:A family F of subsets of [n] is called s-saturated
if it contains no s pairwise disjoint sets\, and
moreover\, no set can be added to F while preservi
ng this property. More than 40 years ago\, Erdős a
nd Kleitman conjectured that an s-saturated family
of subsets of [n] has size at least (1 - 2^-(s-1)
^)2^n^. It is easy to show that every s-saturated
family has size at least 2^n-1^\, but\, as was men
tioned by Frankl and Tokushige\, even obtaining a
slightly better bound of (1/2 + ε)2^n^\, for some
fixed ε > 0\, seems difficult. We prove such a res
ult\, showing that every s-saturated family of sub
sets of [n] has size at least (1 - 1/s)2^n^. This
is joint work with S. Letzter\, B. Sudakov and T.
Tran.\n
LOCATION:MR12
CONTACT:Andrew Thomason
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