# R(5,5;4) > 33

# R(4,5;3) > 32

The three links on this page
point to files that describe a construction related to
R(5,5;4), the smallest non-trivial Ramsey number
for 4-graphs. Using a simple form of simulated annealing I have
produced a K_{5}-free
two-coloring of the edges of K_{33},
thereby proving that R(5,5;4) > 33. Since
there are 40,920 edges in the 4-graph K_{33}, presenting the
construction is somewhat difficult.
So I have provided three files that
may be useful to anyone who wishes to verify the claim.

The file **EdgeList** contains the list of edges which receive
color 0 in my coloring.
The file **AdjArray** contains the "upper simplex" of the
four-dimensional adjacency array.
And the file **Verify.c** is a C program that can read AdjArray and count
monochromatic K_{5}'s, and can be used to verify the claim.
Verify.c reads from stdin.
It may be interesting to note that the program which produced this
construction succeeds very quickly, but is completely stymied when
used to produce a 34-vertex example.
For R(4,5;3) the upper simplex of the adjacancy array is in
this file.

Geoff Exoo