A Tverberg type theorem for matroids

TL;DR

This paper proves a Tverberg-type theorem for matroids, establishing conditions under which multiple disjoint independent sets have intersecting images under continuous maps into Euclidean space, based on the matroid's structure.

Contribution

It introduces a new Tverberg-type theorem for matroids relating disjoint bases to intersection properties under continuous mappings.

Findings

Existence of t disjoint independent sets with intersecting images

t is at least rom the number of disjoint bases in the matroid

The result generalizes classical Tverberg theorems to matroidal complexes.

Abstract

Let b(M) denote the maximal number of disjoint bases in a matroid M. It is shown that if M is a matroid of rank d+1, then for any continuous map f from the matroidal complex M into the d-dimensional Euclidean space there exist t \geq \sqrt{b(M)}/4 disjoint independent sets \sigma_1,\ldots,\sigma_t \in M such that \bigcap_{i=1}^t f(\sigma_i) \neq \emptyset.