User blog comment:Vel!/Call for math facts/@comment-5529393-20140730092437/@comment-5529393-20140730094645

Let R(k,n,c) = the smallest number m such that for any coloring of the subsets of {1,2,...,m} of size k using c colors, there is a subset of size n all of whose subsets of size k are the same color.

Then R(k,n,c) <= (cl)^^k, but not much less. (more than f(c,l)^^(k-1) for some reasonable function of c and l)