Erdős Problem #1176 — Edge Colorings and Monochromatic Vertex Classes

Let $G$ be a graph with chromatic cardinal $\aleph_1$. Is there an edge-coloring of $G$ using exactly $\aleph_1$ colors such that, for every vertex-coloring using at most countably many colors, some vertex-color class contains an edge of every edge color? More precisely, does there exist an edge-color set $E$ of cardinality $\aleph_1$ and a coloring $c_{\mathrm{edge}}$ of the edges of $G$ by $E$ such that, for every vertex-color set $V_c$ with cardinality at most $\aleph_0$ and every vertex-coloring $c_{\mathrm{vert}}$, there is a color $v\in V_c$ for which, for every edge color $e\in E$, some adjacent vertices $u,w$ satisfy $c_{\mathrm{vert}}(u)=c_{\mathrm{vert}}(w)=v$ and $c_{\mathrm{edge}}(\{u,w\})=e$?