Helly Family - Examples

Examples

• In the family of all subsets of the set {a,b,c,d}, the subfamily {{a,b,c}, {a,b,d}, {a,c,d}, {b,c,d}} 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 {a,b,c,d} is a Helly family of order 4.

• Let I be a set of closed intervals of the real line with an empty intersection. Then there must be two intervals A and B in I such that the left endpoint of A is larger than the right endpoint of B. {A,B} have an empty intersection, so I cannot be minimal unless I={A,B}. Therefore, all minimal families of intervals with empty intersections have 2 or fewer intervals in them, and the set of all intervals is a Helly family of order 2.