## New bounds for Szemer\'edi's theorem, III: A polylogarithmic bound for $r_4(N)$

*Green, Ben*;

*Tao, Terence*;

- Subject: Mathematics - Combinatorics | 11B30

Define $r_4(N)$ to be the largest cardinality of a set $A \subset \{1,\dots,N\}$ which does not contain four elements in arithmetic progression. In 1998 Gowers proved that \[ r_4(N) \ll N(\log \log N)^{-c}\] for some absolute constant $c>0$. In 2005, the authors improve... View more

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