2021/09/07 by Minki Kim, Kim, Minki, Alan Lew +1
Computer Science · #Advanced Graph Theory Research #Combinatorics (math.CO) #Data Management and Algorithms #FOS: Mathematics #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2109.03030
openalex publication_date 2021/09/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let K be a simplicial complex on vertex set V. K is called d-Leray if the homology groups of any induced subcomplex of K are trivial in dimensions d and higher. K is called d-collapsible if it can be reduced to the void complex by sequentially removing a simplex of size at most d that is contained in a unique maximal face. We define the t-tolerance complex of K, Tt(K), as the simplicial complex on vertex set V whose simplices are formed as the union of a simplex in K and a set of size at most t. We prove that for any d and t there exists a positive integer h(t,d) such that, for every d-collapsible complex K, the t-tolerance complex Tt(K) is h(t,d)-Leray. The definition of the complex Tt(K) is motivated by results of Montejano and Oliveros on "tolerant" versions of Helly's theorem. As an application, we present some new tolerant versions of the colorful Helly theorem.