Complete graphs in the plane with colored edges where an edge don't cross edges with same color The maximal number of nodes in a complete planar graph is $4$. 
Suppose that the edges of the graph can be chosen with $m$ different colors and that edges with different colors are allowed to cross each other. What would be the maximal number of nodes for a complete graph like this occurring in the plane with this rule?
Down the case with $2$ colors and $7$ nodes.

I found this: layered graphs, and the results there doesn't contradict a conjecture

A complete graph with $4n$ vertices need n colors for the edges.


This isn't a complete answer, but the quantity you're asking for is closely related to the thickness of a a graph, which is the minimum number of planar subgraphs which jointly cover the edges of the graph. It is known that the thickness of the complete graph $K_n$ is $\lfloor (n+7)/6 \rfloor$ except at $K_9$ and $K_{10}$.
I believe that the concepts aren't exactly equivalent, because your framing of the question requires all the planar subgraphs to use the same positions for the vertices, while thickness does not require the same restriction. However, the thickness still gives a lower bound on the number of colors required, and it's possible that diving into the papers in the MathWorld article would show that the constructions involved do use the same positions for all the subgraphs (or could be modified in that way).
