Reasonable Bounds for Combinatorial Lines of Length Three

Abstract

We prove that any subset $A\subseteq [3{]}^n$ with $3^{-n}|A|\ge (\log \log \log \log n{)}^{-c}$ contains a combinatorial line of length $3$, i.e., $x,y,z\in A$, not all equal, with $x_i=y_i=z_i$ or $(x_i,y_i,z_i)=(0,1,2)$ for all $i=1,2,\dots ,n$. This improves on the previous best bound of $3^{-n}|A|\ge \hat{I}\text{c◯}(({\log }^{\ast }n{)}^{-1/2})$ of [D.H.J. Polymath, Ann. of Math. 2012].