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 $\mathrm{\mid }A\mathrm{\mid }>(r-1)(d+1)$. Then, there exist a partition $A_1,\dots ,A_r$ of $A$ and a subset $B\subset A$ of size $(r-1)(d+1)$, such that ${\cap }_{i=1}^r\text{conv}((B\cup \{p\})\cap A_i)\mathrm{\ne }\mathrm{\varnothing }$ 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 $A\setminus B$.