Hypergraphs | Discrete geometry | Families of sets

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. The k-Helly property is the property of being a Helly family of order k. 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 in the family, if , then . 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. (Wikipedia).

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