Frankl's union-closed sets conjecture for families of intermediate height number

Frankl's 1979 union-closed sets conjecture — every finite union-closed family, other than the family consisting only of the empty set, contains an element belonging to at least half of its members — remains open. Tian (2021) introduced the height number $H(F)$ of a union-closed family, the maximum length of a strict inclusion chain of members, proved the conjecture for the end strata $H(F) ≤ 3$ and $H(F) ≥ n−1$ , and proposed the height program: verify the conjecture stratum by stratum over the whole range 1 ≤ $H(F)$ ≤ n, which would prove the conjecture in full. The problem is the program's open middle range 4 ≤ $H(F)$ ≤ n−2. Later literature has moved the frontier but not closed it: the height-three layer was independently peer-reviewed (Colbert, Order 2026), Bouchard (2025) obtained partial averaging results at height four, and Tian's August 2026 preprint claims the complete height-four stratum and reduces a smallest height-five counterexample to three explicit branches (critical join-cover number three or four) while explicitly disclaiming an unrestricted height-five theorem. The surviving open core is the unrestricted empty-set-free height-five case together with all strata up to n−2; completing it, alongside the end strata, would resolve the full conjecture.

Background

A family $F$ of sets is union-closed if $A ∪ B ∈ F$ whenever $A, B ∈ F$ . Frankl's union-closed sets conjecture (1979) asserts that every finite union-closed family other than {∅} contains an element belonging to at least half of its members; such an element is called abundant. In the normalization used by the height program one deletes the empty set — this changes no element frequencies — and asks for an element contained in strictly more than half of the members. Write $U = ∪_{A∈F} A$ for the ground set, $n = |U|$ , $m = |F|$ , and $F_x = \{A ∈ F : x ∈ A\}$ for the subfamily of members containing $x$ (the frequency of $x$ ). The conjecture, open since 1979, is one of the central problems of extremal combinatorics (see the survey of Bruhn and Schaudt). Known partial results fall into three groups. First, structural restrictions on a smallest counterexample: Poonen (1992) showed that a minimal counterexample must be separating — for every pair of distinct ground elements some member contains exactly one of them — initiating the minimal-counterexample method, whose size bounds were pushed further by Norton and Sarvate (1993) and most recently by Hu (2023). Second, information-theoretic bounds: Gilmer (2022) proved that some element lies in a constant fraction (at least 0.01) of the members, a bound improved to (3−√5)/2 ≈ 0.382 by Alweiss, Huang and Sellke (2022) and slightly beyond by Cambie (2022) — still far from the conjectured one half. Third, the conjecture is verified for various special classes (families containing very small sets, very large families, small ground sets).

The height stratification. Order $F$ by inclusion. The height number $H(F)$ , introduced by Tian (2021), is the maximum number of members in a strict inclusion chain $A_1 ⊊ A_2 ⊊ ⋯ ⊊ A_s$ ; equivalently, it is the number of layers obtained by repeatedly deleting the inclusion-minimal members of the family (the height decomposition of the source paper, whose layers the author calls tents when analyzing overlap counts). Since member sizes strictly increase along a chain, 1 ≤ $H(F)$ ≤ n. Tian (2021) proved the conjecture for $H(F) ≤ 3$ and for $H(F) ≥ n−1$ , and proposed the height program: settle the conjecture for each fixed height value; since the heights exhaust 1, …, n, this would prove the conjecture in full. The strata 4 ≤ $H(F)$ ≤ n−2 — the intermediate heights — are the unresolved part and the subject of this problem. Conventions must be tracked carefully: adjoining ∅ to an empty-set-free family leaves all frequencies unchanged while $m$ and the height each increase by one, so the empty-set-free strict-majority statement is equivalent to the standard at-least-half statement for the family with ∅ adjoined; mixing the two normalizations produces spurious one-unit contradictions in the critical even-size case.

Recent progress on the strata. The height-three layer was independently re-proved, peer-reviewed, and extended to infinite families by Colbert (Order, 2026) via optimal elements under chain conditions. Bouchard (2025) proved that a separating union-closed family of height four whose small-set subfamily admits an irredundant cover of size at most two has average member size at least n/2, showed that height four is the largest height for which such an averaging statement can hold, and left cover sizes three and four open. Tian's August 2026 preprint claims the full height-four stratum and, for the next stratum, a conditional structure theorem: a smallest empty-set-free counterexample of height at most five would have even size 2t, at least three critical elements (elements of frequency exactly t), the bound t ≥ 2n−1, the coatom $U ∖ {x}$ (a maximal proper member) present for every critical $x$ , and critical join-cover number — the least number of join-irreducible members (members not expressible as a union of two strictly smaller members) whose union covers all critical elements — equal to three or four. The author explicitly states that no unrestricted height-five theorem is claimed. The open instances that remain are the unrestricted empty-set-free height-five case — reduced to three explicit branches (critical join-cover number three; critical join-cover number four with a full-union cover; critical join-cover number four with the stable two-fiber normal form) and the stable-pair defect transfer inequality — and the strata above it up to n−2, on which no work exists.

Problem Statement

Prove, or refute by an explicit example, the intermediate-height strata of Frankl's union-closed sets conjecture: for every finite union-closed family $F$ of nonempty sets (union-closed meaning $A ∪ B ∈ F$ whenever $A, B ∈ F$ ), with ground set $U = ∪_{A∈F} A$ , $n = |U| ≥ 6$ , and height number $H(F)$ — the maximum number of members in a strict inclusion chain $A_1 ⊊ A_2 ⊊ ⋯ ⊊ A_s$ of $F$ — satisfying 4 ≤ $H(F)$ ≤ n−2, there exists an element $x ∈ U$ contained in strictly more than half of the members of $F$ ; equivalently, some element lies in at least half of the members of the family $F ∪ \{∅\}$ . Together with the end strata $H(F) ≤ 3$ and $H(F) ≥ n−1$ , this completes the height stratification and would establish the full conjecture. A refutation consists of a finite union-closed family of intermediate height in which every element of the ground set is contained in at most half of the members.

References

Chenxiao Tian, Union-closed Sets Conjecture Holds for Height No More Than 3 and Height No Less Than N-1, arXiv:2112.06659 (v1 December 2021, v2 April 2022), doi:10.48550/arXiv.2112.06659, https://arxiv.org/abs/2112.06659 (unrefereed preprint)
Chenxiao Tian, Frankl's Conjecture at Height Four and the Structure of Height-Five Counterexamples, arXiv:2608.25147 (25 August 2026), doi:10.48550/arXiv.2608.25147, https://arxiv.org/abs/2608.25147 (unrefereed preprint)
Cory H. Colbert, Chain Conditions and Optimal Elements in Generalized Union-Closed Families of Sets, Order 43 (published online December 2025; volume year 2026), doi:10.1007/s11083-025-09717-w, https://doi.org/10.1007/s11083-025-09717-w (also arXiv:2412.18740, 2024)
Christopher Bouchard, An averaging result for union-closed families of sets, arXiv:2509.12537 (2025), doi:10.48550/arXiv.2509.12537, https://arxiv.org/abs/2509.12537
Christopher Bouchard, Supersaturation in union-closed families of sets, arXiv:2607.26425 (2026), doi:10.48550/arXiv.2607.26425, https://arxiv.org/abs/2607.26425
Christopher Bouchard, On the lattice formulation of the union-closed sets conjecture, arXiv:2503.00277 (2025), doi:10.48550/arXiv.2503.00277, https://arxiv.org/abs/2503.00277
Justin Gilmer, A constant lower bound for the union-closed sets conjecture, arXiv:2211.09055 (2022), doi:10.48550/arXiv.2211.09055, https://arxiv.org/abs/2211.09055
Ryan Alweiss, Brice Huang, Mark Sellke, Improved Lower Bound for Frankl's Union-Closed Sets Conjecture, arXiv:2211.11731 (2022), doi:10.48550/arXiv.2211.11731, https://arxiv.org/abs/2211.11731
Stijn Cambie, Better bounds for the union-closed sets conjecture using the entropy approach, arXiv:2212.12500 (2022), doi:10.48550/arXiv.2212.12500, https://arxiv.org/abs/2212.12500
Stijn Cambie, Progress on the union-closed conjecture and offsprings in winter 2022-2023, arXiv:2306.12351 (2023), doi:10.48550/arXiv.2306.12351, https://arxiv.org/abs/2306.12351
Henning Bruhn, Oliver Schaudt, The Journey of the Union-Closed Sets Conjecture, Graphs and Combinatorics 31 (2015) 2043–2074, doi:10.1007/s00373-014-1515-0, https://doi.org/10.1007/s00373-014-1515-0
Bjorn Poonen, Union-closed families, Journal of Combinatorial Theory, Series A 59 (1992) 253–268, doi:10.1016/0097-3165(92)90068-6, https://doi.org/10.1016/0097-3165(92)90068-6
Yining Hu, On the union-closed sets conjecture, Discrete Mathematics 346 (2023) 113430, doi:10.1016/j.disc.2023.113430, https://doi.org/10.1016/j.disc.2023.113430
R. M. Norton, D. G. Sarvate, A note of the union-closed sets conjecture, Journal of the Australian Mathematical Society, Series A 55 (1993) 411–413, doi:10.1017/S1446788700034133, https://doi.org/10.1017/S1446788700034133
Roberto Demontis, The union-closed set conjecture is true, arXiv:2405.03731 (2024), doi:10.48550/arXiv.2405.03731, https://arxiv.org/abs/2405.03731 (claimed proof not accepted by the community; cited only as evidence of the conjecture's unresolved status)