A Note on the Tolerant Tverberg Theorem

The tolerant Tverberg theorem generalizes Tverberg’s theorem by introducing a new parameter t called tolerance. It states that there is a minimal number N so that any set of at least N points in \(\mathbb R^d\) can be partitioned into r disjoint sets such that they remain intersecting even after removing any t points from X. In this paper we give an asymptotically tight bound for the tolerant Tverberg Theorem when the dimension and the size of the partition are fixed. To achieve this, we study certain partitions of order-type homogeneous sets and use a generalization of the Erdős–Szekeres theorem. As far as we know, this is the first time that a Ramsey-type theorem has been used to prove a Tverberg-type result.