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Inspired by the Croot-Lev-Pach breakthrough, Jordan Ellenberg and DionGijswijt have recently amazed the combinatorial world by proving that thelargest size of a subset of $F_3^n$ with no 3-term arithmetic progressions isexponentially 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 beautifulproof, that aims to make it appreciated by a wider audience.

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 $c=\root 3 \of {(5589+891\,\sqrt  {33})}/8=2.75510461302363300022127...$