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Source. Published p. 362, Theorem 28 (published scan).

Statement. If each finite configuration $K_i$ is $\ell_i$-Ramsey, then $K_1\times\cdots\times K_t$ is $\ell_1\cdots\ell_t$-Ramsey. Products use mutually orthogonal coordinate spaces. The theorem is about the number of colors used by a forced copy, not a fixed number of available colors.

Complete proof. It suffices to treat two factors and iterate. Fix the available number of colors $r$. By compactness, choose a finite witness $A_2$ such that every $r^{|K_1|}$-coloring of $A_2$ contains a copy of $K_2$ using at most $\ell_2$ colors. Then choose a finite witness $A_1$ such that every $r^{|A_2|}$-coloring of $A_1$ contains a copy of $K_1$ using at most $\ell_1$ colors. This order avoids any circular dependence of the witnesses.

An $r$-coloring $c$ of the orthogonal product $A_1\times A_2$ gives each $x\in A_1$ its row vector $(c(x,y))_{y\in A_2}$. There are at most $r^{|A_2|}$ row types. Choose a congruent $K_1'\subseteq A_1$ with at most $\ell_1$ row types. Next color each $y\in A_2$ by its column restricted to $K_1'$, namely $(c(x,y))_{x\in K_1'}$. There are at most $r^{|K_1|}$ possible columns, so choose a congruent $K_2'\subseteq A_2$ with at most $\ell_2$ column types.

On $K_1'\times K_2'$, points in the same row type and column type have the same original color: moving within a row type preserves the entry at any fixed column, and moving within a column type preserves the entry at any fixed row. Thus there are at most $\ell_1\ell_2$ original colors. Pairwise squared distances add between orthogonal factors, so this is the required congruent product copy. Embed the finite witness product into a sufficiently large Euclidean space, restrict arbitrary ambient colorings to it, and iterate the two-factor argument. $\square$

The source leaves infinite-factor extensions as a separate historical question; no such extension is proved here.

Bears on. #174.