Let $G=(V,E)$ be a simple, undirected graph such that every vertex has degree at least $2$. Given $n\in\mathbb{N}$, a map $c:E \to \{1,\ldots, n\}$ is said to be a *weak coloring* if for every $v\in V$ the edges adjacent to $v$ do not all have the same color. (More formally, we want the restriction $c|_{E(v)}$ to be non-constant, where $E(v) = \{e\in E: x\in e\}$.)

[These][1] [two][2] nice posts by [Mikail Tikhomirov][3] and [Brendan McKay][4] respectively show that for every finite graph there is a weak edge coloring with $3$ colors. I tried to carry through their arguments with transfinite induction to infinite graphs - without success.

**Question.** If $G=(V,E)$ is an infinite simple undirected graph, is there  a weak edge coloring $c:E \to \{1,2,3\}$?

[1]: https://mathoverflow.net/a/285233/8628
[2]: https://mathoverflow.net/a/285234/8628
[3]: https://mathoverflow.net/users/106512/mikhail-tikhomirov
[4]: https://mathoverflow.net/users/9025/brendan-mckay