8 Oct 2026
Jeremy Alm
The Most Wanted Folkman Graph Is Still At Large
Abstract: Any red-blue coloring of the edges of $K_6$, the complete graph on six vertices, must contain a monochromatic triangle. As a corollary, any red-blue edge coloring of a graph $G$ with clique number at least six must contain a monochromatic triangle as well.
One may ask if "six" is strictly necessary in the above statement. In other words, are there graphs with clique number smaller than six, that still have the forced-monochromatic-triangle property? (Spoiler: Yes, there are!) We will explore such graphs and stumble all the way to a stubborn open problem. Bellicose debate will follow.
