Tverberg type theorems for matroids

TL;DR

This paper extends Tverberg's theorem to matroids, providing a new combinatorial partition result with stronger conditions and broader applicability beyond Euclidean spaces.

Contribution

It introduces a matroid-based variant of colorful Tverberg's theorem with enhanced conditions and applicability to any matroid, strengthening previous geometric results.

Findings

Valid in any matroid with finite rank

Partition into rainbow subsequences with nested closures

Applications to non-embeddability results in topology

Abstract

In this paper we show a variant of colorful Tverberg's theorem which is valid in any matroid: Let $S$ be a sequence of non-loops in a matroid $M$ of finite rank $m$ with closure operator cl. Suppose that $S$ is colored in such a way that the first color does not appear more than $r$-times and each other color appears at most $(r-1)$-times. Then $S$ can be partitioned into $r$ rainbow subsequences $S_1,\ldots, S_r$ such that $cl\,\emptyset\subsetneq cl\,S_1\subseteq cl\, S_2\subseteq \ldots \subseteq cl\,S_r$. In particular, $\emptyset\neq \bigcap_{i=1}^r cl\,S_i$. A subsequence is called rainbow if it contains each color at most once. The conclusion of our theorem is weaker than the conclusion of the original Tverberg's theorem in $\mathbb R^d$, which states that $\bigcap conv\,S_i\neq \emptyset$, whereas we only claim that $\bigcap aff\,S_i\neq \emptyset$. On the other hand, our theorem strengthens the Tverberg's theorem in several other ways: 1) it is applicable to any matroid (whereas Tverberg's theorem can only be used in $\mathbb R^d$), 2) instead of $\bigcap cl\,S_i\neq \emptyset$ we have the stronger condition $cl\,\emptyset\subsetneq cl\,S_1\subseteq cl\,S_2\subseteq \ldots \subseteq cl\,S_r$, and 3) we add a color constraints that are even stronger than the color constraints in the colorful version of Tverberg's theorem. Recently, the author together with Goaoc, Mabillard, Pat\'akov\'a, Tancer and Wagner used the first property and applied the non-colorful version of this theorem to homology groups with $GF(p)$ coefficients to obtain several non-embeddability results, for details we refer to arXiv:1610.09063.