Helly-type theorems and fractional Helly-type theorems

A simplicial complex is d-representable if it is the nerve of a collection of convex sets in Rd. Classical Helly's Theorem states that if a d-representable complex contains all the possible faces of dimension d then it is already a full simplex. Helly's Theorem has many extensions and we give a brief survey of some of them. The class of d-representable complexes is a subclass of d-collapsible complexes, and the latter is a subclass of d-Leray complexes. For d 1 we give an example of complexes that are 2d-Leray but not (3d 1)-collapsible. For d 2 we give an example of complexes that are d-Leray but not (2d 2)-representable. We show that for d 3 the complexes from the last example are also d-collapsible. We also give a simple proof of the Combinatorial Alexander Duality, which is a useful topological tool for combinatorics, e.g., for topological versions of Helly's Theorem