A Motivated Rendition of the EllenbergGijswijt Gorgeous proof that the Largest Subset of F_{3}^{n} with No
ThreeTerm Arithmetic Progression is O(c^{n}), with c=2.75510461302363300022127...
By Doron Zeilberger
.pdf
.ps
.tex
(Exclusively published in the Personal Journal of Shalosh B. Ekhad and Doron Zeilberger and arxiv.org )
Written: July 6, 2016
Inspired by the
CrootLevPach
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 3term arithmetic progressions is exponentially less than
3^{n} (and more generally, for F_{q}^{n}, q^{n}),
Here we give a motivated, topdown, 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 F_{q}^{n} with no 3term arithematic progressions.
Sample Input and Output Files for the Maple package Fqn.txt

If you want to see the sequences, recurrence, and asymptotics for F_{3}^{n}
the input file generates the
output file.

If you want to see the sequences, recurrence, and asymptotics for F_{4}^{n}
the input file generates the
output file.

If you want to see the sequences, recurrence, and asymptotics for F_{5}^{n}
the input file generates the
output file.

If you want to see the sequences, recurrence, and asympototics for F_{7}^{n}
the input file generates the
output file.

If you want to see the sequences, recurrence, and asympototics for F_{8}^{n}
the input file generates the
output file.

If you want to see the sequences, recurrence, and asympototics for F_{9}^{n}
the input file generates the
output file.

If you want to see the sequences, recurrence, and asympototics for F_{11}^{n}
the input file generates the
output file.

If you want to see the sequences, recurrence, and asympototics for F_{13}^{n}
the input file generates the
output file.
Personal Journal of Shalosh B. Ekhad and Doron Zeilberger
Doron Zeilberger's Home Page