TITLE: Infinite cuboids In space
QUESTION [8 upvotes]: 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.

REPLY [13 votes]: 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.