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SUMMARY:Increasing Sequences of Integer Triples - Jason Long (University o
 f Cambridge)
DTSTART:20161027T133000Z
DTEND:20161027T143000Z
UID:TALK68662@talks.cam.ac.uk
CONTACT:Andrew Thomason
DESCRIPTION:We will consider the following deceptively simple question\, f
 ormulated recently by Po Shen Loh who connected it to an open problem in R
 amsey Theory. Define the '2-less than' relation on the set of triples of i
 ntegers by saying that a triple x is 2-less than a triple y if x is less t
 han y in at least two coordinates. What is the maximal length of a sequenc
 e of triples taking values in {1\,...\,n} which is totally ordered by the 
 '2-less than' relation?\n\nIn his paper\, Loh uses the triangle removal le
 mma to improve on the trivial upper bound of n^2^ by a factor of log*(n)\,
  and conjectures that the truth should be of order n^(3/2). The gap betwee
 n these bounds has proved to be surprisingly resistant. We shall discuss j
 oint work with Tim Gowers\, giving some developments towards this conjectu
 re and a wide array of natural extensions of the problem. Many of these ex
 tensions remain open.\n
LOCATION:MR12
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