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 $G=(V,E)$. Clearly, $G$ has cliques of cardinality $\aleph_0$. Is $\chi(G) = \aleph_0$?

Let $V$ be the set of all functions $f:\mathbb{N}\to\mathbb{N}$. Let two distinct functions $f,g:\mathbb{N}\to\mathbb{N}$ form an edge if and only if they differ in exactly one input $n\in\mathbb{N}$.

Let $G=(V,E)$. Clearly, $G$ has cliques of cardinality $\aleph_0$. Is $\chi(G) = \aleph_0$?