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 K5-free two-coloring of the edges of K33, thereby proving that R(5,5;4) > 33. Since there are 40,920 edges in the 4-graph K33, 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 K5'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