Let $q\geq 1$ and $H_1,\dots, H_q$ be graphs.

By Ramsey theorem, it is well-known that there exists $n_0$ such that the following holds.

If $n\geq n_0$ and the edges of $K_n$ are colored with $q$ colors, then there exists an $i\in\{1,\ldots,q\}$ such that there exists a monochromatic copy of $H_i$ in color $i$.

The following quantitative version also holds.

There exist positive constants $c>0$ and $n_0$ such that, if $n\geq n_0$ and the edges of $K_n$ are colored with $q$ colors, then there exists an $i\in\{1,\ldots,q\}$ such that there are at least $c n^{|V(H_i)|}$ monochromatic copies of $H_i$ in color $i$.

It is often quoted as "Folklore", and I know it is not too difficult to prove. I've found some related theorem (often related to Ramsey-multiplicities, e.g, this article by Burr and Rosta), but not covering the multicolor / asymmetric case. For completeness, I would like to insert an actual reference in an article.