A Pedagogical Approach to Ramsey Multiplicity

Abstract

It is well known that for all 2-colorings of the edges of $K_6$ there is amonochromatic triangle. Less well known is that there are two monochromatic triangles. More generally, for all 2-colorings of the edges of $K_n$ there are roughly $\ge n^3/24$ monochromatic triangles. Another way to state this is that the density of monochromatic triangles is at least $1/4$. The Ramsey Multiplicity of $k$ is (asymptotically) the greatest $\alpha$ such that for every coloring of $K_n$ the density of monochromatic $K_k$'s is $\alpha$. This concept has been studied for many years. We survey the area and provide proofs that are more complete, more motivated, and using modern notation.