The following question was asked in a comment by Joel David Hamkins in Graph on the set of all functions $f:\mathbb{N}\to\mathbb{N}$.

Let $V$ be the set of all functions $f:\mathbb{N}\to\mathbb{N}$. Let $$E:=\big\{\{f,g\}: f, g \in V \land f\neq g\land \exists k\in \mathbb{N} \text{ }\forall n\in\mathbb{N}\setminus\{k\} (f(n) = g(n))\big\}.$$

Let $G=(V,E)$. This graph is Borel. Can there be a Borel $\mathbb{N}$-coloring?

The following question was asked in a comment by Joel David Hamkins in Graph on the set of all functions $f:\mathbb{N}\to\mathbb{N}$.

Let $V$ be the set of all functions $f:\mathbb{N}\to\mathbb{N}$. Let $$E:=\big\{\{f,g\}: f, g \in V \land f\neq g\land \exists k\in \mathbb{N} \text{ }\forall n\in\mathbb{N}\setminus\{k\} (f(n) = g(n))\big\}.$$

Let $G=(V,E)$. This graph is Borel. Can there be a Borel $\mathbb{N}$-coloring?