The classical Ramsey number R(r_1,...,r_k) is the least n > 0 such that there is no k-coloring of the edges of K_n which does not contain any monochromatic complete subgraph K_ri in color i, for all 1 <= i <= k. In the multicolor case (k > 2). the only known nontrivial value is R(3,3,3) = 17. The only other case whose evaluation does not look hopeless is R(3,3,4), which currently is known to be equal to 30 or 31 by an earlier work of the authors. We report on progress towards deciding which of these tow is the correct value. Using computer algorithms we show that any critical coloring of K_30 proving R(3,3,4) = 31 must satisfy some additional properties, beyond those implied directly by the definitions, further pruning the search space. This progress, though substantial, is not yet sufficient to launch the final attack on the exact value of R(3,3,4).
Department, Program, or Center
Center for Advancing the Study of CyberInfrastructure
Congressus Numerantium 148 (2001) 161-167
RIT – Main Campus