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{{short description|Family of sets where every disjoint subfamily has k or fewer sets}}
In [[combinatorics]], a '''Helly family of order ''k''''' is a family of sets in which every minimal subfamily with an empty intersection has ''k'' or fewer sets in it. Equivalently, every finite subfamily such that every <math>k</math>-fold intersection is non-empty has non-empty total intersection.<ref name="b86">{{citation|title=Combinatorics: Set Systems, Hypergraphs, Families of Vectors, and Combinatorial Probability|first=Béla|last=Bollobás|authorlink=Béla Bollobás|publisher=Cambridge University Press|year=1986|isbn=9780521337038|page=82|url=https://books.google.com/books?id=psqFNlngZDcC&pg=PA82}}.</ref> The ''k''-'''Helly property''' is the property of being a Helly family of order ''k''.<ref name="d95">{{citation▼
▲In [[combinatorics]], a '''Helly family''' of order
| last = Duchet | first = Pierre
| editor1-last = Graham | editor1-first = R. L.
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| publisher = Elsevier
| title = Handbook of combinatorics, Vol. 1, 2
| year = 1995}}. See in particular Section 2.5, "Helly Property", [https://books.google.com/books?id=5Y9NCwlx63IC&pg=PA393 pp. 393–394].</ref>
The number
These concepts are named after [[Eduard Helly]] (1884–1943); [[Helly's theorem]] on [[convex set]]s, which gave rise to this notion, states that convex sets in [[Euclidean space]] of dimension
== Examples ==
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== Formal definition ==
More formally, a '''Helly family of order ''k''''' is a [[set system]] (''V'', ''E''), with ''E'' a collection of [[subset]]s of ''V'', such that, for every finite ''G'' ⊆ ''E'' 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><ref name="b86"/>
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If a family of nonempty sets has an empty intersection, its Helly number must be at least two, so the smallest ''k'' for which the ''k''-Helly property is nontrivial is ''k'' = 2. The 2-Helly property is also known as the '''Helly property'''. A 2-Helly family is also known as a '''Helly family'''.<ref name="b86"/><ref name="d95"/>
A [[convex metric|convex]] [[metric space]] in which the closed [[Ball (mathematics)|balls]] have the 2-Helly property (that is, a space with Helly dimension 1, in the stronger variant of Helly dimension for infinite subcollections) is called [[injective metric space|injective]] or hyperconvex.<ref>{{citation|title=Encyclopedia of Distances|first1=Michel Marie|last1=Deza|author1-link=Michel Deza|first2=Elena|last2=Deza|author2-link=Elena Deza|publisher=Springer|year=2012|isbn=9783642309588|page=19|url=https://books.google.com/books?id=QxX2CX5OVMsC&pg=PA19}}</ref> The existence of the [[tight span]] allows any metric space to be embedded isometrically into a space with Helly dimension 1.<ref>{{citation
| last = Isbell | first = J. R. | authorlink = John R. Isbell
| doi = 10.1007/BF02566944
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== The Helly property in hypergraphs ==
A [[hypergraph]] is equivalent to a set-family. In hypergraphs terms, a hypergraph ''H'' = (''V'', ''E'') has the '''Helly property''' if for every ''n'' hyperedges <math>e_1,\ldots,e_n</math> in ''E'', if <math>\forall i,j\in[n]: e_i \cap e_j \neq\emptyset </math>, then <math>e_1 \cap \cdots \cap e_n \neq\emptyset </math>.<ref name="lp">{{Cite Lovasz Plummer}}</ref>{{Rp|467}} For every hypergraph H, the following are equivalent:<ref name="lp" />{{Rp|
* ''H'' has the Helly property, and the intersection graph of ''H'' (the simple graph in which the vertices are ''E'' and two elements of ''E'' are linked iff they intersect) is a [[perfect graph]].
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{{reflist}}
[[Category:
[[Category:Hypergraphs]]
[[Category:Discrete geometry]]
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