Explicit Construction of Small Folkman Graphs

Abstract

A Folkman graph is a $K_4$-free graph G such that if the edges of G are 2-colored, then there exists a monochromatic triangle. Erdős offered a prize for proving the existence of a Folkman graph with at most 1 million vertices. In this paper, we construct several “small” Folkman graphs within this limit. In particular, there exists a Folkman graph on 9697 vertices.