For any simple, undirected graphs $G, H$, let $G\times H$ denote their [category-theoretical product](https://en.wikipedia.org/wiki/Tensor_product_of_graphs).


What is an example of an infinite connected graph $G$ with $G \cong G \times G$?


(Note that the totally disconnected graph $G = (\omega, \emptyset)$ has $G \cong G \times G$.)



Example of a connected graph $G$ with $G \cong G \times G$
The category of graphs and homomorphisms has countable powers. Given a graph $H = (V(H), E(H))$, its countable power is the graph $H^\omega = (V(H^\omega), E(V^\omega))$ where $V(H^\omega) = V(H)^\omega$ is the countable power of vertices, and $E(H^\omega)$ is defined for $a, b \in V(H)^\omega$ by
$$\{a, b\} \in E(H^\omega) \iff
\forall i \in \omega \,.\, \{a\_i, b\_i\} \in E(H).
$$
For each $i \in \omega$ there is a projection $\pi\_i : H^\omega \to H$, defined by $\pi\_i(a) = a\_i$.
With these projections we really get a power:
$\mathrm{Hom}(L, H^\omega) \cong \mathrm{Hom}(L, H)^\omega$
holds in virtue of the isomorphism taking $f : L \to H^\omega$ to $(\pi\_i \circ f)\_{i \in \omega}$.


For any graph $H$ we have $H^\omega \cong H^\omega \times H^\omega$, with isomorphism taking $a \in V(H^\omega)$ to $((a\_{2 i})\_{i \in \omega}, (a\_{2 j + 1})\_{j \in \omega})$.


Now consider the graph $G = K\_3^\omega$ where $K\_3$ is the complete graph on three vertices $V(K\_3) = \{0, 1, 2\}$. Given $a, b \in \{0,1,2\}^\omega$, we have
$$\{a, b\} \in E(K\_3^\omega) \iff \forall i \in \omega \,.\, a\_i \neq b\_i.$$
The only question remaining is whether $G$ is connected. Take arbitrary vertices $a, b \in \{0,1,2\}^\omega$ and let $c \in \{0,1,2\}^\omega$
be defined by $c\_i = \min (\{0, 1, 2\} \setminus \{a\_i, b\_i\})$. Then $c\_i \neq a\_i$ and $c\_i \neq b\_i$, for all $i \in \omega$, therefore $a$ and $b$ are connected via $c$. The graph $G$ is connected.


