Extensions of the Colorful Helly Theorem for $d$-collapsible and $d$-Leray complexes

Abstract

We present extensions of the Colorful Helly Theorem for $d$-collapsible and $d$-Leray complexes, providing a common generalization to the matroidal versions of the theorem due to Kalai and Meshulam, the ``very colorful" Helly theorem introduced by Arocha, B\'ar\'any, Bracho, Fabila and Montejano, and the ``semi-intersecting" colorful Helly theorem proved by Montejano and Karasev. As an application, we obtain the following extension of Tverberg's Theorem: Let $A$ be a finite set of points in $\mathbb{R}^d$ with $|A|>(r-1)(d+1)$. Then, there exist a partition $A_1,\ldots,A_r$ of $A$ and a subset $B\subset A$ of size $(r-1)(d+1)$, such that $\cap_{i=1}^r \text{conv}( (B\cup\{p\})\cap A_i)\neq\emptyset$ for all $p\in A\setminus B$. That is, we obtain a partition of $A$ into $r$ parts that remains a Tverberg partition even after removing all but one arbitrary point from $A\setminus B$.