A quantitative variant of the multi-colored Motzkin-Rabin theorem

We prove a quantitative version of the multi-colored Motzkin-Rabin theorem in the spirit of [BDWY12]: Let $V_1,ldots,V_n subset R^d$ be $n$ disjoint sets of points (of $n$ `colors). Suppose that for every $V_i$ and every point $v in V_i$ there are at least $delta |V_i|$ other points $u in V_i$ so that the line connecting $v$ and $u$ contains a third point of another color. Then the union of the points in all $n$ sets is contained in a subspace of dimension bounded by a function of $n$ and $delta$ alone.