 Solution

A fixed edge is monochromatic exactly when none of the $r$ hyperplanes separates its endpoints. Each hyperplane fails to cut it with probability at most $1/3$, and the hyperplanes are independent. ThereforeLet $B={\sum }_{ij\in E}{1}_{\{c(i)=c(j)\}}$ count monochromatic edges. Applying linearity of expectation to these indicator random variables givesIndependence between different edges is unnecessary; only the independently sampled hyperplanes are used. The $r$ signs provide at most $2^r$ colors, whether or not every sign region is nonempty.