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CATEGORIES:Combinatorics Seminar
SUMMARY:Cross-intersecting families - Peter Borg (Universi
ty of Malta)
DTSTART;TZID=Europe/London:20180503T143000
DTEND;TZID=Europe/London:20180503T153000
UID:TALK101353AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/101353
DESCRIPTION:A typical problem in extremal set theory is to det
ermine how small or how large a parameter of a sys
tem of sets can be. The Erd\\H{o}s--Ko--Rado~Theor
em is a classical result in this field. A variant
of the Erd\\H{o}s--Ko--Rado problem is to determin
e the maximum sum or the maximum\nproduct of sizes
of k cross-t-intersecting subfamilies $\\mathcal{
A}_1\, \\mathcal{A}_2\, \\dots\, \\mathcal{A}_k$ o
f a given family $\\mathcal{F}$ of sets\, where by
`cross-$t$-intersecting' we mean that\, for every
\n$i$ and $j$ with $i \\neq j$\, each set in $\\ma
thcal{A}_i$ intersects each set in $\\mathcal{A}_j
$ in at least $t$ elements. This natural problem h
as recently attracted much attention. Solutions ha
ve been obtained for various important families\,
such as power sets\, levels of power sets\, heredi
tary families\, families of permutations\, and fam
ilies of integer sequences. The talk will provide
an outline of these results. It will focus mostly
on the product problem for the family of subsets o
f $\\{1\, 2\, \\dots\, n\\}$ that have at most $r$
elements.\n
LOCATION:MR12
CONTACT:Andrew Thomason
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