Helly family

In combinatorics, a Helly family of order k is a family of sets in which every minimal subfamily with an empty intersection has k or fewer sets in it. Equivalently, every finite subfamily such that every k-fold intersection is non-empty has non-empty total intersection.^[1] The k-Helly property is the property of being a Helly family of order k.^[2]

The number k is frequently omitted from these names in the case that k = 2. Thus, a set-family has the Helly property if, for every n sets ${\displaystyle s_{1},\ldots ,s_{n}}$ in the family, if ${\displaystyle \forall i,j\in [n]:s_{i}\cap s_{j}\neq \emptyset }$, then ${\displaystyle s_{1}\cap \cdots \cap s_{n}\neq \emptyset }$.

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.^[1]