Reasonable Bounds for Combinatorial Lines of Length Three

Amey Bhangale, Subhash Khot, Yang P. Liu +1

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…