A Motivated Rendition of the Ellenberg-Gijswijt Gorgeous proof that the   Largest Subset of $F_3^n$ with No Three-Term Arithmetic Progression is   $O(c^n)$, with $c=\root 3 \of {(5589+891\,\sqrt   {33})}/8=2.75510461302363300022127...$
  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. 
Doron Zeilberger
math.CO