Joonkyung Lee (이준경), On Ramsey multiplicity

Ramsey’s theorem states that, for a fixed graph $H$, every 2-edge-colouring of $K_n$ contains a monochromatic copy of $H$ whenever $n$ is large enough. Perhaps one of the most natural questions after Ramsey’s theorem is then how many copies of monochromatic $H$ can be guaranteed to exist. To formalise this question, let the Ramsey multiplicity $M(H;n)$ be the minimum number of labelled copies of monochromatic $H$ over all 2-edge-colouring of $K_n$. We define the Ramsey multiplicity constant $C(H)$ is defined by $C(H):=\underset{n\to \mathrm{\infty }}{lim}\frac{M(H,n)}{n(n-1)\cdots (n-v+1)}$. I will discuss various bounds for C(H) that are known so far.