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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?

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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.

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