2023/05/21 by Kim, Minki, Lew, Alan
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2305.12360
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 ∩i=1r 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.