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NEW BOUNDS FOR SZEMERÉDI'S THEOREM, III: A POLYLOGARITHMIC BOUND FOR r 4 ( N )
Author(s) -
Green Ben,
Tao Terence
Publication year - 2017
Publication title -
mathematika
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.955
H-Index - 29
eISSN - 2041-7942
pISSN - 0025-5793
DOI - 10.1112/s0025579317000316
Subject(s) - mathematics , cardinality (data modeling) , combinatorics , constant (computer programming) , arithmetic progression , limit (mathematics) , set (abstract data type) , upper and lower bounds , discrete mathematics , arithmetic , mathematical analysis , computer science , data mining , programming language
Definer 4 ( N )to be the largest cardinality of a set A ⊂ { 1 , … , N } that does not contain four elements in arithmetic progression. In 1998, Gowers proved thatr 4 ( N ) ≪ N ( log log N ) − cfor some absolute constant c > 0 . In 2005, the authors improved this tor 4 ( N ) ≪ N e − c log log N.In this paper we further improve this tor 4 ( N ) ≪ N ( log N ) − c ,which appears to be the limit of our methods.