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SUMMARY:Sets without four term progressions but rich in three term progres
 sions - Oliver Roche-Newton (RICAM\, Linz\, Austria)
DTSTART:20190529T124500Z
DTEND:20190529T134500Z
UID:TALK122620@talks.cam.ac.uk
CONTACT:Aled Walker
DESCRIPTION:The main question that will be addressed in this talk is the f
 ollowing: given a set A which does not contain any four term arithmetic pr
 ogressions\, is it necessarily the case that there exists a large subset o
 f A which does not contain any three term arithmetic progressions?\n\nPerh
 aps one might guess that the answer is "yes"\, and that by deleting a rela
 tively small number of elements from A we can destroy all progressions. In
  fact this rough intuition seems to be false\, as we aim to show in this t
 alk by constructing sets (in both the integers and finite field setting) w
 ith no 4APs but for which all large subsets contain a 3AP. Possible connec
 tions with quantitative bounds for Roth's Theorem will also be discussed. 
 The proof uses the method of hypergraph containers.\n\nThis talk is based 
 on joint work with Cosmin Pohoata
LOCATION:MR12\, CMS\, Wilberforce Road\, Cambridge\, CB3 0WB
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