Helly family
In combinatorics, a Helly family of order k is a family of sets such that any minimal subfamily with an empty intersection has k or fewer sets in it. Equivalently, every finite subfamily such that every -fold intersection is non-empty has non-empty total intersection.
The k-Helly property is the property of being a Helly family of order k. These concepts are named after Eduard Helly (1884 - 1943); Helly's theorem on convex sets, which gave rise to this notion, states that convex sets in Euclidean space of dimension n are a Helly family of order n + 1. The number k is frequently omitted from these names in the case that k = 2.
More formally, a Helly family of order k is a set system (F, E), with F a collection of subsets of E, such that, for every finite G ⊆ F with
we can find H ⊆ G such that
and
In some cases, the same definition holds for every subcollection G, regardless of finiteness. However, this is a more restrictive condition. For instance, the open intervals of the real line satisfy the Helly property for finite subcollections, but not for infinite subcollections: the intervals (0,1/i) (for i = 0, 1, 2, ...) have pairwise nonempty intersections, but have an empty overall intersection.
If a family of sets is a Helly family of order k, that family is said to have Helly number k. The Helly dimension of a metric space is one less than the Helly number of the family of metric balls in that space; Helly's theorem implies that the Helly dimension of a Euclidean space equals its dimension as a real vector space.
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