Ramsey Theory says that every sufficently large (but finite) complete graph having $d-$coloured edges contains a monochromatic complete subgraph with $k$ vertices.


One could ask for asymptotics: Let $A(n,d,k)$ be the minimal number of
monochromatic complete subgraphs with $k$ vertices contained in any complete graph with $n$
vertices whose edges are coloured with $d$ colours.


One has the obvious bounds: $n+1-N\leq A(n,d,k)\leq {n\choose k}\sim \frac{n^k}{k!}$
where $N$ is the corresponding Ramsey number.


There exists thus a critical exponent $\alpha$ 
such that $\lim\_{n\rightarrow\infty}\frac{A(n,d,k)}{n^{\beta}}=\infty$
for all $\beta<\alpha$ and 
$\liminf\_{n\rightarrow\infty}\frac{A(n,d,k)}{n^{\beta}}=0$
for all $\beta>\alpha$. 


What is $\alpha$? Perhaps the function $\frac{A(n,d,k)}{n^{\alpha}}$ has a nice behaviour?


Are there any instance where one can say anything interesting? (One has 
obviously $\alpha\geq 1$ a strict inequality would probably already be interesting.)


The same type of question can be asked for Erd\"os-Szekeres, for van der Waerden 
(with a different, much smaller but still affine lower bound) etc.


Added after Gowers solution: Gowers gave the following easy proof that $\alpha=k$.
Denote by $m=\inf\lbrace n\ \vert\ A(n,d,k)>0\rbrace$ the Ramsey number 
corresponding to our problem. Since a fixed monochromatic $k-$clique is contained
in exactly ${n-k\choose m-k}$ subsets of size $m$ we have 
$$A(n,d,k)\geq\frac{{n\choose m}}{{n-k\choose m-k}}\sim \frac{n^k}{m^k}\ .$$


The only interesting question is thus the exact value of $\lim\_{n\rightarrow\infty}\frac{A(n,d,k)}{n^k}$ which exists by Gowers first argument.



Asymptotics for Ramsey Theory
I'm not sure this is as interesting as you think. Here is a sketch that $\alpha=k$. For large enough $n$ we can apply Szemer\'edi's regularity lemma simultaneously to all the colour classes. The result is a collection of block graphs that add up to 1 everywhere (approximately). If the number of blocks is larger than the corresponding Ramsey number (plus a little bit to allow for the error) then you get a block-clique joined by block-edges all of the same colour. Then a counting lemma gives you a number of cliques that is proportional (with a very small constant of proportionality) to $n^k$. 


Actually, I'm being stupid. As I know perfectly well, since I've used it frequently, one can just average over all subgraphs with m vertices, where m is the relevant Ramsey number. Each one contains a monochromatic clique, and since m doesn't depend on n the number of cliques you end up with is proportional to $n^k$.


Essentially the same argument works for van der Waerden and many other problems.


Edit: actually, the regularity lemma comment wasn't entirely stupid, because it can be used to show that $A(n,k,d)/n^k$ tends to a limit as $n$ tends to infinity, answering another of your questions above.


