Helly family
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In combinatorics, a Helly family of order k, named after Eduard Helly (1884 - 1943), is a set system (F, E), with F a collection of subsets of E, that satisfies the k-Helly property. This says that an empty intersection of sets from the family can always be refined to an empty intersection of at most k, for given k ≥ 2.
The 2-Helly property is also known as the Helly property. A 2-Helly family is also known as a Helly family. It is easy to see that the set of intervals on the real line has the Helly property.
More formally, the property of a Helly family of order k is that, for any G ⊆ F with
- <math>\bigcap_{X\in G} X=\varnothing,<math>
we can find H ⊆ G such that
- <math>\bigcap_{X\in H} X=\varnothing<math>
and
- <math>\left|H\right|\le k.<math>
Helly's theorem on convex sets, which rise gave to this notion, states that convex sets in Euclidean space of dimension n are a Helly family of order n + 1.Template:Math-stub