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SUMMARY:Cross-intersecting families - Peter Borg (University of Malta)
DTSTART:20180503T133000Z
DTEND:20180503T143000Z
UID:TALK101353@talks.cam.ac.uk
CONTACT:Andrew Thomason
DESCRIPTION:A typical problem in extremal set theory is to determine how s
 mall or how large a parameter of a system of sets can be. The Erd\\H{o}s--
 Ko--Rado~Theorem is a classical result in this field. A variant of the Erd
 \\H{o}s--Ko--Rado problem is to determine the maximum sum or the maximum\n
 product of sizes of k cross-t-intersecting subfamilies $\\mathcal{A}_1\, \
 \mathcal{A}_2\, \\dots\, \\mathcal{A}_k$ of 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 $\\mathcal{A}_i$ intersects each se
 t in $\\mathcal{A}_j$ in at least $t$ elements. This natural problem has r
 ecently attracted much attention. Solutions have been obtained for various
  important families\, such as power sets\, levels of power sets\, heredita
 ry families\, families of permutations\, and families 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 of $\\{1\, 2\, \\dots\, n
 \\}$ that have at most $r$ elements.\n
LOCATION:MR12
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