thumb|Each hyperedge of a [[hypergraph is a set of vertices. No vertex is shared by all 4 edges. Some subfamilies with 3 edges created by removing an edge are non-intersecting (edges 1, 3 and 4), but others aren't (edges 1, 2 and 3 which all share vertex 3), so the 4 edge family is not minimal. However, it isn't possible to remove an edge from either of the non-intersecting 3 edge subfamilies to get a non-intersecting 2 edge subfamily. Thus, the edges form a Helly family of order 3.]]
In combinatorics, a Helly family of order is a family of sets in which every minimal subfamily with an empty intersection has or fewer sets in it. Equivalently, every finite subfamily such that every -fold intersection is non-empty has non-empty total intersection. The -Helly property is the property of being a Helly family of order .
The number is frequently omitted from these names in the case that . Thus, a set-family has the Helly property if, for every sets <math>s_1,\ldots,s_n</math> in the family, if <math>\forall i,j\in[n]: s_i \cap s_j \neq\emptyset </math>, then <math>s_1 \cap \cdots \cap s_n \neq\emptyset </math>.
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 are a Helly family of order .
- The family of infinite arithmetic progressions of integers 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.
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 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 (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. The existence of the tight span allows any metric space to be embedded isometrically into a space with Helly dimension 1.
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>. For every hypergraph H, the following are equivalent: