Let us denote with $\omega_i \,,\, i=1,2,\ldots,n-1$ the (counterclockwise) rotation about the center of the polygon with angle $\frac{2\pi i}{n}$; $R$ be the set of red points and $B$ - the set of black points. Claim: There exists a rotation $\omega_i$, such that $|\omega_i(R)\cap B| > p/2$. Suppose on the contrary this is not true. Thus, $|\omega_i(R)\cap B| \leq p/2 \,,\, i=1,2,\ldots, n-1$. Therefore: \[ \sum_{i=1}^{n-1} |\omega_i(R)\cap B| \leq \frac{(n-1) p}{2} \] On the other hand: \[ \sum_{i=1}^{n-1}|\omega_i(R)\cap B| = p (n-p) \] It follows: \[ p(n-p) \leq \frac{(n-1) p}{2} \Rightarrow n-p \leq \frac{n-1}{2} \] But according to the hypothesis $n-p >n/2$ , which is a contradiction.