Graphs with Monochromatic Complete Subgraphs in Every Edge Coloring

Abstract

For integers r, $s\geqq 2$, let $\Gamma (r,s)$ be the class of all graphs G with the following property if the edges of G are colored red and blue, then either G contains r mutually adjacent vertices with all connecting lines colored red, or s mutually adjacent vertices with all connecting lines colored blue. By Ramsey's theorem, $\Gamma (r,s)$ contains all sufficiently large complete graphs. It follows that $\Gamma (r,s)$ contains all graphs with a sufficiently large number of mutually adjacent vertices. We are concerned here with determining the minimum number $f = f(r,s)$ such that $\Gamma (r,s)$ contains a graph G with at most f mutually adjacent vertices. Obviously $f(r,s)\geqq \max (r,s)$. We show constructively that $f(r,s) = \max (r,s)$.