Erdős Problem #1128 — Monochromatic Countable Boxes in Triple Colorings

For sets $A_1\subseteq A$, $B_1\subseteq B$, and $C_1\subseteq C$, a box $A_1\times B_1\times C_1$ is monochromatic under a coloring $f:A\to B\to C\to\operatorname{Fin}2$ if there exists $c\in\operatorname{Fin}2$ such that $f(a,b,c')=c$ for every $a\in A_1$, every $b\in B_1$, and every $c'\in C_1$. Is it true that for all sets $A,B,C$ with $|A|=|B|=|C|=\aleph_1$, and every coloring $f:A\to B\to C\to\operatorname{Fin}2$, there exist subsets $A_1\subseteq A$, $B_1\subseteq B$, and $C_1\subseteq C$ with $|A_1|=|B_1|=|C_1|=\aleph_0$ such that $A_1\times B_1\times C_1$ is monochromatic?