Problem #638: Let $S$ be a family of finite graphs such that for every...

Let $S$ be a family of finite graphs such that for every $n$ there is some $G_n\in S$ such that if the edges of $G_n$ are coloured with $n$ colours then there is a monochromatic triangle.

Is it true that for every infinite cardinal $\aleph$ there is a graph $G$ of which every finite subgraph is in $S$ and if the edges of $G$ are coloured with $\aleph$ many colours then there is a monochromatic triangle.