Quantitative Ramsey theorem - asymmetric and multicolors

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.

• This immediately follows from the usual Ramsey theorem by averaging over subsets of size $N$=Ramsey number. So most probably it was not published as a separate result, and many people got it independently. Oct 23 '20 at 12:04
• @FedorPetrov yep that's my guess. I know the proof, I could write a summary of it. I was just hoping for some reference. Thanks Oct 24 '20 at 2:54