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, 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 Rd with |A| > (r − 1)(d + 1). Then, there exist a partition A1, . . ., Ar of A and a subset B ⊂ A of size (r − 1)(d + 1) such that ∩ri=1 conv((B ∪ {p}) ∩ Ai) ≠ ∅ for all p ∈ A \ 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 \ B. © The Author(s), 2024.