New bounds for the same-type lemma

Boris Bukh, Alexey Vasileuski

Given finite sets $X_1,\dots ,X_m$ in ${\mathbb{R}}^d$ (with $d$ fixed), we prove that there are respective subsets $Y_1,\dots ,Y_m$ with $|Y_i|\ge \frac{1}{\operatorname{poly}(m)}|X…