Title: Tverberg type theorems for matroids,
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, 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.