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SUMMARY:Constructions of Turán systems that are tight up to a multiplicat
 ive constant - Oleg Pikhurko (Warwick)
DTSTART:20260319T143000Z
DTEND:20260319T153000Z
UID:TALK245230@talks.cam.ac.uk
CONTACT:103978
DESCRIPTION:The Turán density t(s\,r) is the asymptotically smallest edge
  density\nof an r-graph for which every set of s vertices contains at leas
 t one\nedge. The question of estimating this function received a lot of\na
 ttention over decades of attempts. A trivial lower bound is t(s\,r)\\ge\n1
 /{s\\choose s−r). In the early 1990s\, de Caen conjectured that\nt(r+1\,
 r) grows faster than O(1/r) and offered 500 Canadian dollars for\nresolvin
 g this question.\n\nI will give an overview of this area and present a con
 struction\ndisproving this conjecture by showing more strongly that for ev
 ery\ninteger R there is C such that t(r+R\,r)\\le C/{r+R\\choose R}\, that
  is\,\nthe trivial lower bound is tight for every fixed R up to a\nmultipl
 icative constant C=C(R).
LOCATION:MR12
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