What I want to show is that among $2^p+1$ points in the plane there are three that determine an angle of size at least $\pi(1-1/p)$.
I was told I have to start with showing for $n=2^p$ that the graph $K_{n+1}$ is not the union of $p$ bipartite graphs but $K_n$ is.
I have no idea how to prove this. My first ideas were induction or proof by contradition but neither was helpful. It could also have something to do with the fact that every bipartite grpah is 2-colourable.