Extensions of the colorful Helly theorem for d-collapsible and d-Leray complexes

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, Bárány, 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 \operatorname {\mathrm {\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 $A\setminus B$.