by Aaron Robertson and Doron Zeilberger
Written: Feb. 27, 1998.
Issai Schur proved the first Ramsey Theorem (way before
Ramsey). He proved that if you color the integers
from 1 to n by r-colors, you are GUARANTEED (if n is
not too small) a monochromatic triple {x,y,x+y}.
[By the way, his motivation was to prove
Fermat's Last Theorem.]
If you hate Schur triples, you must learn to
put up with them. Nevertheless, you may still want
to 2-color in such a way as to MINIMIZE
these buggers.
If the number of colors is 2, Ron Graham, in
SOCA 96' (Tianjin, June 1996), offered
100 dollars for the asymptotic minimal number of
these Schur triples.
This is yet another example of an
a posteriori trivial theorem. It is highly non-trivial
to CONJECTURE the statements that lead to the
solution, using the Maple package
RON,
but the difficulty is that of
experimental math , not `rigorous' math.
Once the right statements are conjectured, the
formal proof is ROUTINE, and can be safely
left to the obtuse reader.
Tomasz Schoen, a student of Tomasz Luczak, has independently
solved this problem.
.pdf
.ps
.tex
Appears in
Electronic Journal of Combinatorics, v. 5(1998), R19.
Doron Zeilberger's List of Papers