Helly Family

Helly Family

In combinatorics, a Helly family of order k is a family of sets such that any minimal subfamily with an empty intersection has k or fewer sets in it. In other words, any subfamily such that every -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. 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 n are a Helly family of order n + 1.