Helly numbers of acyclic families

Abstract : The Helly number of a family of sets with empty intersection is the size of its largest inclusion-wise 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 also prove a stronger theorem where the intersection of 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 is better than what was previously known. In fact, our proof bounds the Leray number of the nerves of the families under consideration and thus also yields, under similar assumptions, a fractional Helly theorem, a (p,q)-theorem and the existence of small weak ϵ-nets.
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Éric Colin De Verdière, Grégory Ginot, Xavier Goaoc. Helly numbers of acyclic families. Advances in Mathematics, Elsevier, 2014, 253, pp.163-193. 〈10.1016/j.aim.2013.11.004〉. 〈hal-00646166〉

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