Leck–Roberts–Simpson weighted union-closure conjecture

Establish that for every positive integers N and k, every family A of N distinct k-subsets, and every nonnegative nondecreasing sequence of weights (w_j), the union-closure weight satisfies w(⟨A⟩) ≥ w(⟨F_k(N)⟩), where F_k(N) consists of the first N k-subsets in max-lexicographic order.

Background

The paper studies the weighted union-closure minimization problem in which each set of cardinality j receives a nonnegative weight w_j, with the weights nondecreasing in j. The conjectured extremal family is generated by the first N k-subsets in max-lexicographic order.

The paper proves substantial threshold-weight special cases, including the layered conjecture in the first strip and several additional ranges. However, the general weighted assertion is not established; the authors note that the layered formulation over all cutoffs is equivalent to the weighted conjecture, so unresolved layered cases remain unresolved instances of this broader problem.