Tverberg partitions as weak epsilon-nets

Abstract: We prove a Tverberg-type theorem using the probabilistic method. Given $\varepsilon >0$, we find the smallest number of partitions of a set $X$ in $R^d$ into $r$ parts needed in order to induce at least one Tverberg partition on every subset of $X$ with at least $\varepsilon |X|$ elements. This generalizes known results about Tverberg's theorem with tolerance.