Helly numbers of acyclic families

The Helly number of a family of sets with empty intersection is the size of its largest inclusionwise minimal sub-family with empty intersection. Let F be a finite family of open subsets of an arbitrary locally arc-wise connected topological space Γ. Assume that for every sub-family G ⊆ F the intersection of the elements of G has at most r connected components, each of which is a Q-homology cell. We show that the Helly number of F is at most r(dΓ + 1), where dΓ is the smallest integer j such that every open set of Γ has trivial Q-homology in dimension j and higher. (In particular dRd = d). This bound is best possible. We prove, in fact, a stronger theorem where small sub-families may have more than r connected components, each possibly with nontrivial homology in low dimension. As an application, we obtain several explicit bounds on Helly numbers in geometric transversal theory for which only ad hoc geometric proofs were previously known; in certain cases, the bound we obtain i...

• Type: Journal
• Year: 2011
• Where: CORR
• Authors: Éric Colin de Verdière, Grégory Ginot, Xavier Goaoc
• Keywords: Connected Topological Space, CORR 2011, Education, Helly Numbers, Inclusionwise Minimal Sub-family