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On the complex plane $\mathbb C$ consider the half-open square $$\square=\{z\in\mathbb C:0\le\Re(z)<1,\;0\le\Im(z)<1\}.$$

Observe that for every $z\in \mathbb C$ and $r\in\{0,1,2,3\}$ the set $(z+i^r\cdot\square)$ is the shifted and rotated square $\square$ with a vertex at $z$.

Problem. Is it true that for any function $r:\mathbb C\to\{0,1,2,3\}$ there a subset $Z\subset\mathbb C$ such that the union of the squares $$\bigcup_{z\in Z}(z+i^{r(z)}\cdot\square)$$is not Borel in $\mathbb C$?

Added in Edit. As @YCor observed in his comment, the answer to this problem is affirmative under $\neg CH$.

An affirmative answer to Problem would follow from an affirmative answer to another intriguing

Problem'. Is it true that for any partition $\mathbb C=A\cup B$ either $A$ contains an uncountable strictly increasing function or $B$ contains an uncountable strictly decreasing function?

Here by a function I understand a subset $f\subset \mathbb R\times\mathbb R=\mathbb C$ such that for any $x\in\mathbb R$ the set $f(x)=\{y\in\mathbb R:(x,y)\in f\}$ contains at most one element.

Added in the Next Edit. In the discussion with @YCor we came to the conclusion that under CH the answer to both problems is negative. Therefore, both problems are independent of ZFC. Very strange.

On the complex plane $\mathbb C$ consider the half-open square $$\square=\{z\in\mathbb C:0\le\Re(z)<1,\;0\le\Im(z)<1\}.$$

Observe that for every $z\in \mathbb C$ and $p\in\{0,1,2,3\}$ the set $(z+i^p\cdot\square)$ is the shifted and rotated square $\square$ with a vertex at $z$.

Problem. Is it true that for any function $p:\mathbb C\to\{0,1,2,3\}$ there a subset $Z\subset\mathbb C$ such that the union of the squares $$\bigcup_{z\in Z}(z+i^{p(z)}\cdot\square)$$is not Borel in $\mathbb C$?

Added in Edit. As @YCor observed in his comment, the answer to this problem is affirmative under $\neg CH$.

An affirmative answer to Problem would follow from an affirmative answer to another intriguing

Problem'. Is it true that for any partition $\mathbb C=A\cup B$ either $A$ contains an uncountable strictly increasing function or $B$ contains an uncountable strictly decreasing function?

Here by a function I understand a subset $f\subset \mathbb C$ such that for any $x\in\mathbb R$ the set $f(x)=\{y\in\mathbb R:x+iy\in f\}$ contains at most one element.

Added in the Next Edit. In the discussion with @YCor we came to the conclusion that under CH the answer to both problems is negative. Therefore, both problems are independent of ZFC. Very strange.

On the complex plane $\mathbb C$ consider the half-open square $$\square=\{z\in\mathbb C:0\le\Re(z)<1,\;0\le\Im(z)<1\}.$$

Observe that for every $z\in \mathbb C$ and $r\in\{0,1,2,3\}$ the set $(z+i^r\cdot\square)$ is the shifted and rotated square $\square$ with a vertex at $z$.

Problem. Is it true that for any function $r:\mathbb C\to\{0,1,2,3\}$ there a subset $Z\subset\mathbb C$ such that the union of the squares $$\bigcup_{z\in Z}(z+i^{r(z)}\cdot\square)$$is not Borel in $\mathbb C$?

Added in Edit. As @YCor observed in his comment, the answer to this problem is affirmative under $\neg CH$.

An affirmative answer to Problem would follow from an affirmative answer to another intriguing

Problem'. Is it true that for any partition $\mathbb C=A\cup B$ either $A$ contains an uncountable strictly increasing function or $B$ contains an uncountable strictly decreasing function?

Here by a function I understand a subset $f\subset \mathbb R\times\mathbb R=\mathbb C$ such that for any $x\in\mathbb R$ the set $f(x)=\{y\in\mathbb R:(x,y)\in f\}$ contains at most one element.

Added in the Next Edit. In the discussion with @YCor we came to the conclusion that under CH the answer to both problems is negative. Therefore, both problems are independent of ZFC. Very strange.

On the complex plane $\mathbb C$ consider the half-open square $$\square=\{z\in\mathbb C:0\le\Re(z)<1,\;0\le\Im(z)<1\}.$$

Observe that for every $z\in \mathbb C$ and $r\in\{0,1,2,3\}$ the set $(z+i^r\cdot\square)$ is the shifted and rotated square $\square$ with a vertex at $z$.

Problem. Is it true that for any function $r:\mathbb C\to\{0,1,2,3\}$ there a subset $Z\subset\mathbb C$ such that the union of the squares $$\bigcup_{z\in Z}(z+i^{r(z)}\cdot\square)$$is not Borel in $\mathbb C$?

Added in Edit. As @YCor observed in his comment, the answer to this problem is affirmative under $\neg CH$.

An affirmative answer to Problem would follow from an affirmative answer to another intriguing

Problem'. Is it true that for any partition $\mathbb C=A\cup B$ either $A$ contains an uncountable strictly increasing function or $B$ contains an uncountable strictly decreasing function?

Here by a function I understand a subset $f\subset \mathbb R\times\mathbb R=\mathbb C$ such that for any $x\in\mathbb R$ the set $f(x)=\{y\in\mathbb R:(x,y)\in f\}$ contains at most one element.

Added in the Next Edit. In the discussion with @YCor we came to the conclusion that under CH the answer to both problems is negative. Therefore, both problems are independent of ZFC. Very strange.

On the complex plane $\mathbb C$ consider the half-open square $$\square=\{z\in\mathbb C:0\le\Re(z)<1,\;0\le\Im(z)<1\}.$$

Observe that for every $z\in \mathbb C$ and $r\in\{0,1,2,3\}$ the set $(z+i^r\cdot\square)$ is the shifted and rotated square $\square$ with a vertex at $z$.

Problem. Is it true that for any function $r:\mathbb C\to\{0,1,2,3\}$ there a subset $Z\subset\mathbb C$ such that the union of the squares $$\bigcup_{z\in Z}(z+i^{r(z)}\cdot\square)$$is not Borel in $\mathbb C$?

Added in Edit. As @YCor observed in his comment, the answer to this problem is affirmative under $\neg CH$.

An affirmative answer to Problem would follow from an affirmative answer to another intriguing

Problem'. Is it true that for any partition $\mathbb C=A\cup B$ either $A$ contains an uncountable strictly increasing function or $B$ contains an uncountable strictly decreasing function?

Here by a function I understand a subset $f\subset \mathbb R\times\mathbb R=\mathbb C$ such that for any $x\in\mathbb R$ the set $f(x)=\{y\in\mathbb R:(x,y)\in f\}$ contains at most one element.

Added in the Next Edit. In the discussion with @YCor we came to the conclusion that under CH the answer to both problems in negative. Therefore, both problems are independent of ZFC. Very strange.

On the complex plane $\mathbb C$ consider the half-open square $$\square=\{z\in\mathbb C:0\le\Re(z)<1,\;0\le\Im(z)<1\}.$$

Observe that for every $z\in \mathbb C$ and $r\in\{0,1,2,3\}$ the set $(z+i^r\cdot\square)$ is the shifted and rotated square $\square$ with a vertex at $z$.

Problem. Is it true that for any function $r:\mathbb C\to\{0,1,2,3\}$ there a subset $Z\subset\mathbb C$ such that the union of the squares $$\bigcup_{z\in Z}(z+i^{r(z)}\cdot\square)$$is not Borel in $\mathbb C$?

Added in Edit. As @YCor observed in his comment, the answer to this problem is affirmative under $\neg CH$.

An affirmative answer to Problem would follow from an affirmative answer to another intriguing

Problem'. Is it true that for any partition $\mathbb C=A\cup B$ either $A$ contains an uncountable strictly increasing function or $B$ contains an uncountable strictly decreasing function?

Here by a function I understand a subset $f\subset \mathbb R\times\mathbb R=\mathbb C$ such that for any $x\in\mathbb R$ the set $f(x)=\{y\in\mathbb R:(x,y)\in f\}$ contains at most one element.

Added in the Next Edit. In the discussion with @YCor we came to the conclusion that under CH the answer to both problems is negative. Therefore, both problems are independent of ZFC. Very strange.