Non-Monochromatic Triangles in a 2-Edge-Coloured Graph

Abstract

Let $G = (V,E)$ be a simple graph and let $\{R,B\}$ be a partition of $E$.  We prove that whenever $|E| + \mathrm{min}\{ |R|, |B| \} > { |V| \choose 2 }$, there exists a subgraph of $G$ isomorphic to $K_3$ which contains edges from both $R$ and $B$. If instead equality holds, and $G$ has no such subgraph, then we show that $G$ is in one of a few simple classes.