论文信息 - A topological colorful Helly theorem

A topological colorful Helly theorem

Abstract Let F 1 ,…, F d+1 be d+1 families of convex sets in R d . The Colorful Helly Theorem (see (Discrete Math. 40 (1982) 141)) asserts that if ⋂ i=1 d+1 F i ≠∅ for all choices of F 1 ∈ F 1 ,…,F d+1 ∈ F d+1 then there exists an 1⩽i⩽d+1 such that ⋂ F∈ F i F≠∅ . Our main result is both a topological and a matroidal extension of the colorful Helly theorem. A simplicial complex X is d-Leray if H i (Y; Q )=0 for all induced subcomplexes Y⊂X and i⩾d. Theorem.LetXbe ad-Leray complex on the vertex setV. Suppose M is a matroidal complex on the same vertex setVwith rank functionρ. IfM⊂Xthen there exists a simplexτ∈Xsuch thatρ(V−τ)⩽d.

Gil Kalai | Roy Meshulam | G. Kalai | R. Meshulam

[1] Roy Meshulam,et al. The Clique Complex and Hypergraph Matching , 2001, Comb..

[2] A. Björner. Topological methods , 1996 .

[3] Ron Aharoni,et al. Hall's theorem for hypergraphs , 2000 .

[4] K. S. Sarkaria. Tverberg’s theorem via number fields , 1992 .

[5] R. Graham,et al. Handbook of Combinatorics , 1995 .

[6] Noga Alon,et al. Point Selections and Weak ε-Nets for Convex Hulls , 1992, Combinatorics, Probability and Computing.

[7] Roy Meshulam,et al. Domination numbers and homology , 2003, J. Comb. Theory A.

[8] Jörg M. Wills,et al. Handbook of Convex Geometry , 1993 .

[9] Imre Bárány,et al. A generalization of carathéodory's theorem , 1982, Discret. Math..

[10] G. Wegner,et al. d-Collapsing and nerves of families of convex sets , 1975 .

[11] J. Eckhoff. Helly, Radon, and Carathéodory Type Theorems , 1993 .