Fix positive integers N,k,n with n≥ k. Let mathfrak F_N,k be the class of finite union-closed families F of finite sets satisfying |A∈ F:|A|=k|=N. Order the k-element subsets of N first by their largest element and, when these are equal, lexicographically by their increasing lists. Let S_N,k be the first N sets in this order, and define U_N,k=bigcup_A∈ GA:varnothing≠ G⊆ S_N,k. The conjecture asserts that, for every n≥ k, min_ F∈mathfrak F_N,k|A∈ F:|A|≥ n|=|A∈ U_N,k:|A|≥ n|. Thus, the family of all nonempty unions of the first N k-subsets is a minimizer for the number of members of size at least n among finite union-closed families containing exactly N distinct k-sets.

Leck–Roberts–Simpson weighted conjecture

Fix positive integers $N,k,n$ with $n\ge k$ . Let $\mathfrak F_{N,k}$ be the class of finite union-closed families $\mathcal F$ of finite sets satisfying $|\{A\in\mathcal F:|A|=k\}|=N$ . Order the $k$ -element subsets of $\mathbb N$ first by their largest element and, when these are equal, lexicographically by their increasing lists. Let $\mathcal S_{N,k}$ be the first $N$ sets in this order, and define $\mathcal U_{N,k}=\{\bigcup_{A\in\mathcal G}A:\varnothing\ne\mathcal G\subseteq\mathcal S_{N,k}\}$ . The conjecture asserts that, for every $n\ge k$ , $\min_{\mathcal F\in\mathfrak F_{N,k}}|\{A\in\mathcal F:|A|\ge n\}|=|\{A\in\mathcal U_{N,k}:|A|\ge n\}|$ . Thus, the family of all nonempty unions of the first $N$ $k$ -subsets is a minimizer for the number of members of size at least $n$ among finite union-closed families containing exactly $N$ distinct $k$ -sets.

References

Primary source

arXiv

Additional references

On union-closed families with prescribed number of k-sets — arXiv — Amir Jafari

combinatorics extremal-set-theory

Posted 3 weeks ago · edited 3 weeks ago

Progress summary

Refreshed 3 weeks ago

Claimed progress

A new preprint expands the cases where the conjectured best construction is known to work, but does not settle the conjecture.

The Leck–Roberts–Simpson conjecture predicts that an initial segment minimizes the relevant weighted union-closure quantity. The conjecture remains unresolved beyond the parameter ranges now covered by the latest preprint.

Known results

For $k=2$ , Leck, Roberts, and Simpson proved the exact initial-segment result (2012).
Randelović proved the corresponding large- $t$ unweighted bound and recorded the broader conjectures as open (2023).

September 2026 parameter-range advance

A September 2026 preprint by Amir Jafari reports new strip-by-strip and asymptotic ranges in which the conjectured initial-segment minimizer is valid, including a sufficient ground-set bound of order $k^2t/\log k$ for large $k$ . This is a substantial claimed advance, but it does not cover the full weighted conjecture and has not been independently verified here.

Current status (as of September 2026): The conjecture is proved in the classical $k=2$ case and in the newly reported parameter regimes, while the general weighted case remains open.

Sources

No solutions have been posted yet.