Given an arbitrary $X \subseteq \mathbb{R}^3$, can we always find infinite sets $A, B, C \subseteq \mathbb{R}$ such that either $A \times B \times C \subseteq X$ or $A \times B \times C \subseteq (\mathbb{R}^3 \setminus X)$? Here $\mathbb{R}$ denotes the real line. 


I believe that the answer should be well known (and positive?), but I don't see it.



Infinite cuboids In space
This is Problem 28 in the paper Unsolved problems in set theory of Erdos and Hajnal <https://old.renyi.hu/~p_erdos/1971-28.pdf> (at least if CH holds). In an 1982 paper <https://old.renyi.hu/~p_erdos/1982-24.pdf> Erdos reports that Mills and Prikry solved it negatively, but I could not locate their paper. 


