Given $c<\infty$ colors, positive integers $k_1,\dots,k_n$ and positive integers $N_1,\dots,N_n$. Then there exist positive integers $M_1,\dots,M_n$ so that for disjoint finite sets $A_1,\dots,A_n$ of cardinalities $|A_i|=M_i$, $1\leq i\leq n$, the following statement holds:
assume that each array $(B_1,\dots,B_n)$, where $B_i\subset A_i$ and $|A_i|=k_i$ is colored in one of our $c$ colors. Then there always exist sets $C_i\subset A_i$, $|C_i|=N_i$, so that colors of arrays satisfying $B_i\subset C_i$ are all the same.
This follows from the usual Ramsey theorem by straightforward induction, but maybe it is well known statement itself?