On the most wanted Folkman graph

Stanislaw Radziszowski
Xu Xiaodong

Note: imported from RIT’s Digital Media Library running on DSpace to RIT Scholar Works in February 2013. This article can also be viewed online at: http://www.uccs.edu/~geombina/


We discuss a branch of Ramsey theory concerning edge Folkman numbers. Fe(3; 3; 4) involves the smallest parameters for which the problem is open, posing the question \What is the smallest order N of a K4-free graph, for which any 2-coloring of its edges must contain at least one monochromatic triangle?" This is equivalent to finding the order N of the smallest K4-free graph which is not a union of two triangle-free graphs. It is known that 16 is less than or equal to N (an easy bound), and it is known through a probabilistic proof by Spencer that N is less than or equal to 3 X 10^9. In this paper, after overviewing related Folkman problems, we prove that 19 is less than or equal to N, and give some evidence for the bound N is less than equal to 27.