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SUMMARY:Monotone arrays and a multidimensional Ramsey Theorem -  Gal Krone
 nberg (Oxford)
DTSTART:20250313T143000Z
DTEND:20250313T153000Z
UID:TALK229249@talks.cam.ac.uk
CONTACT:103978
DESCRIPTION:A foundational result in Ramsey theory appears in a paper of E
 rdős and Szekeres from 1935: any sequence of n^2 +1 distinct real numbers
  contains either an increasing or decreasing subsequence of length n+1. Th
 is simple result was one of the starting seeds for the development of Rams
 ey theory. We discuss a generalisation of the Erdős-Szekeres theorem to m
 onotone arrays. We will show how to obtain improvements on a theorem prove
 d by Fishburn and Graham 30 years ago thus confirming a conjecture posed b
 y Bucic\, Sudakov\, and Tran. More precisely\, we will show that a doubly 
 exponential upper bound holds in all dimensions. Finally\, we will see how
  this is intimately connected to a generalisation of Ramsey Theorem on the
  cartesian product of cliques. \nJoint work with Antonio Girao and Alex Sc
 ott.\n
LOCATION:MR12
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