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New bounds for Szemeredi's theorem. III: A polylogarithmic bound for r_4(N)

Abstract:
Define r4(N) to be the largest cardinality of a set A ⊂ {1, . . . , N} which does not contain four elements in arithmetic progression. In 1998 Gowers proved that r4(N) L N(log log N) -c for some absolute constant c > 0. In 2005, the authors improved this to r4(N) L Ne-c v log log N . In this paper we further improve this to r4(N) L N(log N) -c , which appears to be the limit of our methods.
Publication status:
Published
Peer review status:
Peer reviewed
Version:
Accepted Manuscript

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Publisher copy:
10.1112/S0025579317000316

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Department:
Magdalen College
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Funding agency for:
Green, BJ
Publisher:
London Mathematical Society Publisher's website
Journal:
Mathematika Journal website
Volume:
63
Issue:
3
Pages:
944-1040
Publication date:
2017-11-29
Acceptance date:
2017-08-07
DOI:
EISSN:
2041-7942
ISSN:
0025-5793
Pubs id:
pubs:713166
URN:
uri:1d09eef3-01e2-4ce0-ab9d-2892019812c8
UUID:
uuid:1d09eef3-01e2-4ce0-ab9d-2892019812c8
Local pid:
pubs:713166

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