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 Barak et al. (Proceedings of the National Academy of Sciences, 2012): Let $$V_1,\ldots ,V_n \subset {\mathbb {R}}^d$$V1,?,Vn?Rd be $$n$$n disjoint sets of points (of $$n$$n `colors'). Suppose that for every $$V_i$$Vi and every point $$v \in V_i$$v?Vi there are at least $$\delta |V_i|$$?|Vi| other points $$u \in V_i$$u?Vi so that the line connecting $$v$$v and $$u$$u contains a third point of another color. Then the union of the points in all $$n$$n sets is contained in a subspace of dimension bounded by a function of $$n$$n and $$\delta $$? alone.