A Motivated Rendition of the Ellenberg-Gijswijt Gorgeous proof that the Largest Subset of F3n with No Three-Term Arithmetic Progression is O(cn), with c=2.75510461302363300022127...

By Doron Zeilberger

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(Exclusively published in the Personal Journal of Shalosh B. Ekhad and Doron Zeilberger and arxiv.org )

Written: July 6, 2016

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 F3n with no 3-term arithmetic progressions is exponentially less than 3n (and more generally, for Fqn, qn), Here we give a motivated, top-down, rendition of their beautiful proof, that aims to make it appreciated by a wider audience.

Maple package

• Fqn.txt, a Maple package to compute the sequences (and their asympototics) that feature as upper bounds for the size of subsets of Fqn with no 3-term arithematic progressions.

Sample Input and Output Files for the Maple package Fqn.txt

• If you want to see the sequences, recurrence, and asymptotics for F3n

the input file generates the output file.

• If you want to see the sequences, recurrence, and asymptotics for F4n

the input file generates the output file.

• If you want to see the sequences, recurrence, and asymptotics for F5n

the input file generates the output file.

• If you want to see the sequences, recurrence, and asympototics for F7n

the input file generates the output file.

• If you want to see the sequences, recurrence, and asympototics for F8n

the input file generates the output file.

• If you want to see the sequences, recurrence, and asympototics for F9n

the input file generates the output file.

• If you want to see the sequences, recurrence, and asympototics for F11n

the input file generates the output file.

• If you want to see the sequences, recurrence, and asympototics for F13n

the input file generates the output file.