Tverberg type theorems for matroids
Pavel Paták · arXiv (Cornell University) · 2017
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, $\emptyset eq \bigcap_{i=1}^r cl\,S_i$. 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 $\bigcap conv\,S_i eq \emptyset$, whereas we only claim that $\bigcap aff\,S_i eq \emptyset$. 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 $\bigcap cl\,S_i eq \emptyset$ 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, Patáková, 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.