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? 



Multipartite Ramsey theorem
This is indeed well known, although I can't give a reference where it is stated in this exact form. If I see right, Theorem 49 of Erdos-Rado: A partition calculus in set theory, Bull. Amer. Math. Soc. 62 (1956), 427--489 (available as <http://www.renyi.hu/~p_erdos/1956-02.pdf>) is at least very close to it. 
R. Rado: Direct decompositions of partitions, Journal of Lon. Math. Soc 29(1954), 71-83. may contain the theorem, but I don't have access to it. See J.Larson: Infinite Combinatorics, Handbook of the History of Logic, vol 6, p. 215. 


