About the piercing number of a family of intervals

Given a universe (a set) U and a property P, (closed under inclusions, for subsets of U). Results of the type “if every subset of cardinality μ of a finite family F ⊂ U has property P, then the entire family F has property P”are called Helly type theorems. The minimum number μ for which the result is true is called the Helly number of the Helly-type theorem (U ,P , μ). In the case of Helly’s classical theorem, U is the family of convex sets in the euclidean space R, the property P is to have a point in common and μ = d+ 1.