A note on the tolerated Tverberg theorem

TL;DR

This paper establishes an asymptotically tight bound for the tolerated Tverberg Theorem in fixed dimensions and partition sizes, utilizing partitions of order-type homogeneous sets and a generalized Erdős–Szekeres theorem.

Contribution

It provides the first asymptotically tight bound for the tolerated Tverberg Theorem in fixed dimension and partition size, advancing combinatorial geometry.

Findings

Derived an asymptotically tight bound for the tolerated Tverberg Theorem

Analyzed partitions of order-type homogeneous sets

Applied a generalized Erdős–Szekeres theorem

Abstract

In this paper we give an asymptotically tight bound for the tolerated 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 Erd\H{o}s-Szekeres theorem.