Layered union-closure conjecture for prescribed numbers of k-sets

Determine whether every union-closed family F containing exactly N k-sets satisfies |F_{≥ n}| ≥ |⟨F_k(N)⟩_{≥ n}| for all positive integers N, k, n with n ≥ k, where F_k(N) is the family of the first N k-subsets in max-lexicographic order.

Background

The central extremal question asks whether the max-lexicographic generators minimize the number of generated sets whose sizes meet a prescribed cutoff n. The conjecture is formulated for arbitrary union-closed families containing exactly N k-sets and is equivalent, over all cutoffs, to the Leck–Roberts–Simpson weighted conjecture.

The paper resolves the conjecture in the first strip, in the second strip for triples, on minimum support in several higher-strip regimes, and for sufficiently large cutoffs relative to k and the strip index. It does not establish the assertion for all N, k, and n.