This year we celebrate 50 years of the lovely theorem of Tverberg!

Let $a_{1},\ldots,a_{n}$ be points in $\mathbb{R}^{d}$. If the number of points $n$ satisfies $n >(d+1)(m-1)$, then they can be partitioned into $m$ disjoint parts $A_{1},\ldots,A_{m}$ in such a way that the $m$ convex hulls $\operatorname{conv} A_1, \ldots, \operatorname{conv} A_m$ have a point in common.

Over the years many generalizations and extensions, including colorful, fractional, and topological versions, have been developed and are a bounty for discrete geometers. My talk will introduce yet another fascinating way to interpret these theorems, now with a view toward Number theory in mind. Given a discrete set $S$ of $\mathbb{R}^d$ (e.g., a lattice, or the Cartesian product of the prime numbers), we study the number of points of $S$ needed to guarantee the existence of an $m$-partition of the points $A_{1},\ldots,A_{m}$ such that the intersection of the $m$ convex hulls of the parts contains at least $k$ points of $S$. The proofs of the main results require new quantitative versions of Helly's and Caratheodory's theorems. Joint work with Reuben La Haye, David Rolnick, and Pablo Soberon.