A Quantitative Variant of the Multi-colored Motzkin–Rabin Theorem

Abstract

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 $V_1,\ldots ,V_n \subset {\mathbb {R}}^d$ 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.