Tverberg theorems over discrete sets of points

This paper discusses Tverberg-type theorems with coordinate constraints (i.e., versions of these theorems where all points lie within a subset $S \subset \mathbb{R}^d$ and the intersection of convex hulls is required to have a non-empty intersection with $S$). We determine the $m$-Tverberg number, when $m \geq 3$, of any discrete subset $S$ of $\mathbb{R}^2$ (a generalization of an unpublished result of J.-P. Doignon). We also present improvements on the upper bounds for the Tverberg numbers of $\mathbb{Z}^3$ and $\mathbb{Z}^j \times \mathbb{R}^k$ and an integer version of the well-known positive-fraction selection lemma of J. Pach.