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In
combinatorics Combinatorics is an area of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geom ...
, 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 k-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''.. See in particular Section 2.5, "Helly Property"
pp. 393–394
The number ''k'' is frequently omitted from these names in the case that ''k'' = 2. Thus, a set-family has the Helly property if for every ''n'' sets s_1,\ldots,s_n in the family, if \forall i,j\in s_i \cap s_j \neq\emptyset , then s_1 \cap \cdots \cap s_n \neq\emptyset . These concepts are named after
Eduard HellyEduard Helly (June 1, 1884 in Vienna en, Viennese , iso_code = AT-9 , registration_plate = Vehicle registration plates of Austria, W , postal_code_type = Postal code , postal_code ...
(1884–1943);
Helly's theorem Helly's theorem is a basic result in discrete geometry A collection of circles and the corresponding unit disk graph Discrete geometry and combinatorial geometry are branches of geometry that study Combinatorics, combinatorial properties and con ...

Helly's theorem
on
convex set In geometry, a subset of a Euclidean space, or more generally an affine space over the Real number, reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set o ...

convex set
s, which gave rise to this notion, states that convex sets in
Euclidean space Euclidean space is the fundamental space of classical geometry. Originally, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension (mathematics), dimens ...
of dimension ''n'' are a Helly family of order ''n'' + 1.


Examples

* In the family of all subsets of the set , the subfamily has an empty intersection, but removing any set from this subfamily causes it to have a nonempty intersection. Therefore, it is a minimal subfamily with an empty intersection. It has four sets in it, and is the largest possible minimal subfamily with an empty intersection, so the family of all subsets of the set is a Helly family of order 4. * Let ''I'' be a finite set of closed intervals of the
real line In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
with an empty intersection. Let ''A'' be the interval whose left endpoint ''a'' is as large as possible, and let ''B'' be the interval whose right endpoint ''b'' is as small as possible. Then, if ''a'' were less than or equal to ''b'', all numbers in the range 'a'',''b''would belong to all intervals of ''I'', violating the assumption that the intersection of ''I'' is empty, so it must be the case that ''a'' > ''b''. Thus, the two-interval subfamily has an empty intersection, and the family ''I'' cannot be minimal unless ''I'' = . Therefore, all minimal families of intervals with empty intersections have two or fewer intervals in them, showing that the set of all intervals is a Helly family of order 2. * The family of infinite
arithmetic progression An Arithmetic progression (AP) or arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant. For instance, the sequence 5, 7, 9, 11, 13, 15, . . . is an arithmetic progression with a common diffe ...

arithmetic progression
s of
integer An integer (from the Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'', meaning "to share" or "to ...
s also has the 2-Helly property. That is, whenever a finite collection of progressions has the property that no two of them are disjoint, then there exists an integer that belongs to all of them; this is the
Chinese remainder theorem In number theory, the Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer An integer (from the Latin wikt:integer#Latin, ''integer'' meaning "whole") is colloquially defined as a number ...
.


Formal definition

More formally, a Helly family of order ''k'' is a
set systemIn set theory illustrating the intersection (set theory), intersection of two set (mathematics), sets. Set theory is a branch of mathematical logic that studies Set (mathematics), sets, which informally are collections of objects. Although any typ ...
(''V'', ''E''), with ''E'' a collection of
subset In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

subset
s of ''V'', such that, for every finite ''G'' ⊆ ''E'' with :\bigcap_ X=\varnothing, we can find ''H'' ⊆ ''G'' such that :\bigcap_ X=\varnothing and :\left, H\\le k. 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 interval In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
s 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.


Helly dimension

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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...
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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
. The Helly dimension of a subset S of a Euclidean space, such as a polyhedron, is one less than the Helly number of the family of translates of S. For instance, the Helly dimension of any
hypercube In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of ...

hypercube
is 1, even though such a shape may belong to a Euclidean space of much higher dimension. Helly dimension has also been applied to other mathematical objects. For instance defines the Helly dimension of a
group A group is a number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...
(an algebraic structure formed by an invertible and associative binary operation) to be one less than the Helly number of the family of left cosets of the group.


The Helly property

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. A
convex Convex means curving outwards like a sphere, and is the opposite of concave. Convex or convexity may refer to: Science and technology * Convex lens A lens is a transmissive optics, optical device which focuses or disperses a light beam by me ...
metric space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...
in which the closed
balls A ball A ball is a round object (usually spherical, but can sometimes be ovoid An oval (from Latin ''ovum'', "egg") is a closed curve in a plane which resembles the outline of an egg. The term is not very specific, but in some areas ( p ...
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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
or hyperconvex. The existence of the
tight spanImage:Orthogonal-convex-hull.svg, If a set of points in the plane, with the taxicab geometry, Manhattan metric, has a connected orthogonal convex hull, then that hull coincides with the tight span of the points. In metric geometry, the metric envelop ...
allows any metric space to be embedded isometrically into a space with Helly dimension 1..


The Helly property in hypergraphs

A
hypergraph File:PAOH hypergraph representation.png, alt=PAOH visualization of a hypergraph, Alternative representation of the hypergraph reported in the figure above, called PAOH. Edges are vertical lines connecting vertices. V7 is an isolated vertex. Vertice ...

hypergraph
is equivalent to a set-family. In hypergraphs terms, a hypergraph ''H'' = (''V'', ''E'') has the Helly property if for every ''n'' hyperedges e_1,\ldots,e_n in ''E'', if \forall i,j\in e_i \cap e_j \neq\emptyset , then e_1 \cap \cdots \cap e_n \neq\emptyset . For every hypergraph H, the following are equivalent: * ''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 240px, The Paley graph of order 9, colored with three colors and showing a clique of three vertices. In this graph and each of its induced subgraphs the chromatic number equals the clique number, so it is a perfect graph. In graph theory, a perfect ...
. * Every partial hypergraph of ''H'' (i.e., a hypergraph derived from ''H'' by deleting some hyperedges) has the Konig property, i.e., its maximum- matching size equals its minimum- transversal size. * Every partial hypergraph of ''H'' has the property that its maximum degree equals its minimum
edge coloring. In graph theory In mathematics, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''Vertex (graph the ...

edge coloring
number.


References

{{reflist Set families Hypergraphs