  Inspired by the Croot-Lev-Pach breakthrough, Jordan Ellenberg and Dion
Gijswijt have recently amazed the combinatorial world by proving that the
largest size of a subset of $F_3^n$ with no 3-term arithmetic progressions is
exponentially less than the size, $3^n$ of $F_3^n$ (and, more generally, $q^n$
for $F_q^n$). Here we give a motivated, top-down, rendition of their beautiful
proof, that aims to make it appreciated by a wider audience.
