Helly number

$ \def\X{\mathcal X} % family of sets \def\S{\mathcal S} % subfamily $

The Helly number $ H(\X) $ of a family of sets $\X$ is (in analogy to Helly's theorem) the smallest natural number $k$ such that the following (compactness-type) intersection property holds:

Let $ \S $ be a subfamily of $ \X $. If any $k$ members of $\S$ have a common point, then the sets of $\S$ have a common point.

This is also called the Helly property, and the corresponding is called a Helly family (of order $k$).