Consider the following family of sets $$ \begin{align*} \mathcal{F} = \{X\subseteq [0,1]\times [0,1] \mid \ &X \text{ is closed and }\\& \forall x \in \pi_0 (X) (\{y \in [0,1] \mid (x,y) \in X\} \text{ is perfect}) \text{ and}\\& \forall y \in \pi_1 (X) (\{x \in [0,1] \mid (x,y) \in X\} \text{ is perfect})\} \end{align*} $$

where $\pi_0$ and $\pi_1$ are the canonical projections. My question is the following:

- Given any nonempty $X \in \mathcal{F}$ are there two nonempty perfect $C_0,C_1 \subseteq [0,1]$ such that $C_0 \times C_1 \subseteq X$?

Instead of $[0,1]$ I could have put any other nonempty perfect Polish space. I tried to define a counterexample via a construction *à la* Cantor (Cantor sets, Sierpiński carpets etc.), but without success. Any idea? Do you know if these sets have been studied somewhere?

Thanks

EDIT1: Here is a variation of the previous question:

- Given any nonempty $X\in\mathcal{F}$ are there two $C_0,C_1\subseteq [0,1]$ such that both $C_0$ and $C_1$ have at least two elements and $C_0\times C_1 \subseteq X$?

EDIT2: I'm starting to believe that the answer to both questions may be positive and that Ramsey theory may have something to do with it, but at this moment I cannot make this feeling precise.

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