REVIEW 2 major objections 3 minor 2 cited by
Frequent elements in union-closed set families
T0 review · 2 major / 3 minor · reviewed 2026-08-11 · deepseek-v4-flash
Pith's one-line read The paper proves the 2022 conjecture that the kth-most frequent element in any union-closed family appears in at least |F|/(2^{k-1}+1) sets.
desk verdict The entropy half is solid, but a false inequality in Proposition 3.3 leaves a substantial middle range of family sizes unproven; the result is likely fixable but not yet proven. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
For large families, the load-bearing device is an entropy inequality (Lemma 2.5): if a random set $A$ on a union-closed family has every element appearing with probability at most $\alpha < (3-\sqrt{5})/2$, then the union of two independent copies satisfies $H(A \cup B) \ge \lambda_\alpha H(A)$, where $\lambda_\alpha = H(2\alpha-\alpha^2)/H(\alpha) > 1$. To apply this to the kth frequency, the paper projects the family by deleting the $k-1$ most frequent elements through the map $A \mapsto A \setminus [k-1]$, then compares entropies before and after projection. For small families, the central combinatorial object is a minimal k-good set: a set $S$ outside the top $k-1$ elements that meets every family member containing an element outside that range; union-closedness forces $|S| \le \log_2 m$, which yields the claimed frequency bound.
What would settle it
For $k=6$ and $m=14$, the claimed implication reads $14 - 32 > 7$, which is false; this is exactly the line based on '$m > 2k$' in the proof of Proposition 3.3. A concrete search for a union-closed family with 14 sets, support at least 6, and sixth-most frequent element frequency below $1/33$ would decide whether Theorem 1.4 itself is true in this range.
Extended reading notes
Core claim
The main theorem (Theorem 1.4) states that for $k \ge 2$, if a family $F$ is union-closed and $|\cup_{A \in F} A| \ge k$, then the kth-most frequent element lies in at least $|F|/(2^{k-1}+1)$ sets in $F$, with equality only if $F$ is a near-k-cube. Since near-k-cubes satisfy the bound, the result is best possible. The proof splits into three size regimes: very small families by direct inspection, intermediate families by a covering argument, and large families by an entropy inequality applying to elements with frequency below $(3-\sqrt{5})/2$. The same machinery yields Corollary 4.1: for fixed $k$, $f_k(F) \ge (3-\sqrt{5})/2 - o(1)$ as $|F| \to \infty$.
Load-bearing premise
The argument for mid-sized families assumes that if the family has more than $2k$ sets, then removing the sets built from the $k-1$ most popular elements still leaves more than half the sets; this is false when the family has between $2k+2$ and $2^k$ sets.
Editorial extensions
If this is right
- The kth-frequency conjecture holds for every k ≥ 2, with near-k-cubes as the only equality cases.
- For any fixed k, f_k(F) ≥ (3−√5)/2 − o(1) as |F| → ∞.
- In large union-closed families, the kth-most frequent element is asymptotically as common as the most frequent element.
- The k = 1 case is exactly the Union-Closed Sets Conjecture, so the generalisation is settled while the original problem remains open.
Reading between the lines
- A natural next step the paper does not take is a stability version: if the kth frequency is close to |F|/(2^{k−1}+1), the family should be structurally close to a near-k-cube.
- The projection trick that removes the top k−1 elements could be iterated to bound the sum of the k largest frequencies, not just the kth frequency alone.
- If future improvements to the entropy constant for the most frequent element can be made to survive the projection, Corollary 4.1 would improve for every k at once.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies Nagel's generalization of Frankl's union-closed sets conjecture, asserting that the kth-most frequent element of a union-closed family with support at least k occurs in at least |F|/(2^{k-1}+1) sets. The authors split the proof into an entropic regime (Theorem 2.6 and Proposition 2.7) and a small/middle regime (Propositions 3.1 and 3.3), and they also derive an asymptotic corollary f_k(F) >= (3-√5)/2 - o(1) for |F| -> ∞. The core difficulty for families of intermediate size is Proposition 3.3, whose proof contains a false inequality; as a result, the main theorem is not established as written.
Significance. If the result were proved, it would fully resolve a natural and well-motivated generalization of Frankl's conjecture with a clean equality characterization. Section 2 is a solid and substantial contribution: Theorem 2.6 gives explicitly quantified entropic thresholds, Proposition 2.7 settles small k exactly, and Corollary 4.1 is an appealing consequence. Observation 1.3, connecting Nagel's conjecture to the union-closed sets conjecture, is also elegant. The proof is not circular and the entropy argument appears sound; however, the paper's central claim depends crucially on Proposition 3.3, and the false step in that proposition leaves an exponentially large family-size range untreated.
major comments (2)
- [Section 3, proof of Proposition 3.3] The line "since m > 2k, we have m - 2^{k-1} > 1/2 m" is false: it is equivalent to m > 2^k, not m > 2k. The proposition is stated for all m with 2k+2 <= m <= 2^{3(k-1)}, and for every k >= 5 this range contains the interval [2k+2, 2^k]. In that interval the displayed lower bound (m - 2^{k-1})/(m log2 m) is negative near the lower endpoint and is generally far too small to imply f_k(F) > 1/(2^{k-1}+1). Since Proposition 2.7 only applies for m >= 2^{2.71(k-1)} and Proposition 3.1 only for m <= 2k+1, no argument in the paper covers these sizes, so Theorem 1.4 is not established as written.
- [Section 3, k=5 subcase of Proposition 3.3] The same false implication recurs in the k=5 subcase: the assertion "if m >= 2^{17/2}, then we actually have m - 16 > 15/16 m" requires m > 256, whereas 2^{17/2} is approximately 181.02. Thus the claimed bound f_5(F) >= 15/(16 log2 m) >= 5/64 is not valid for 181 <= m <= 256, and this interval is not covered by the surrounding cases.
minor comments (3)
- [Abstract and formatted displays] The fraction 1/(2^{k-1}+1) appears garbled in the abstract and in several inline displays; the rendering should be corrected throughout.
- [Observation 1.3] The phrase "pi_{k-1}^{-1}(F) \subseteq F contains at least one set" is confusing; it should state that the preimage of F under the projection, restricted to the family F, is nonempty.
- [Table 1] In Table 1, the row for Proposition 3.3 should explicitly indicate that it applies only for k >= 5; the current formatting leaves this implicit.
Circularity Check
No circularity: the proof combines external entropic lemmas and independent counting, with no fitted parameters or load-bearing self-citations.
full rationale
The paper's derivation chain is self-contained in the relevant sense. For large families, Theorem 2.6 applies the external entropic result of Sawin (Lemma 2.5) to the projected family, bounds H(A|B) via the support size, and derives an upper bound on |F|; no step defines f_k(F) in terms of the target bound 1/(2^{k-1}+1). For small families, Proposition 3.1 is a direct counting argument, and Proposition 3.3 adapts Knill's combinatorial method to bound a minimal k-good vertex cover, again by counting distinct unions. The three propositions cover disjoint ranges of m and are combined in Section 4 to prove Theorem 1.4. There are no fitted parameters, no self-citations used as evidence, and no imported uniqueness theorem. The correctness issue identified by the skeptic — the inequality 'm > 2k implies m - 2^{k-1} > m/2' being false for k >= 5 in Proposition 3.3 — is a mathematical error in the proof rather than a circularity; per the review rules, a false inference belongs under correctness risk, not circularity analysis. Accordingly, no circular step can be exhibited, and the score is 0.
Assumptions & free parameters
assumptions (3)
- standard math Sawin's entropy inequality (Lemma 2.5) is valid as stated
- standard math The projection π_{k−1}(F) of a union-closed family is union-closed
- standard math Standard entropy properties: chain rule, range, conditioning lowers entropy, data processing
Cite this review
Pith. "Pith review of Frequent elements in union-closed set families." pith.science (2026). https://pith.science/paper/NDUVXQPQ
@misc{pith2026241203862,
author = {Pith},
title = {Pith review of: Frequent elements in union-closed set families},
year = {2026},
howpublished = {\url{https://pith.science/paper/NDUVXQPQ}},
note = {Machine review of arXiv:2412.03862}
}
abstract
The Union-Closed Sets Conjecture asks whether every union-closed set family $\mathcal{F}$ has an element contained in half of its sets. In 2022, Nagel posed a generalisation of this problem, suggesting that the $k$th-most popular element in a union-closed set family must be contained in at least $\frac{1}{2^{k-1} + 1} |\mathcal{F}|$ sets. We combine the entropic method of Gilmer with the combinatorial arguments of Knill to show that this is indeed the case for all $k \ge 2$, and characterise the families that achieve equality. Furthermore, we show that when $|\mathcal{F}| \to \infty$, the $k$th-most frequent element will appear in at least $\left( \frac{3 - \sqrt{5}}{2} - o(1) \right) |\mathcal{F}|$ sets, reflecting the recent progress made for the Union-Closed Set Conjecture.
Forward citations
Cited by 2 Pith papers
-
A lemma on a finite union-closed family of finite sets and its applications
A lemma bounding element frequencies under deletion implies the equivalence of Frankl's conjecture and Nagel's conjecture, and strengthens a bound of Nagel for sets of size at least two.
-
Further analysis on the second frequency of union-closed set families
The paper proves that if a union-closed family has second-most-frequent element frequency at most 1/3, then it must have between 81 and 113 sets and all its minimal 2-good sets have size 4.
Reference graph
Works this paper leans on
-
[1]
James Aaronson, David Ellis, and Imre Leader, A note on transitive union-closed families, Electron. J. Combin. 28 (2021), no. 2, Paper No. 2.3, 4
work page 2021
-
[2]
Ryan Alweiss, Brice Huang, and Mark Sellke, Improved lower bound for F rankl's union-closed sets conjecture , Electron. J. Combin. 31 (2024), no. 3, Paper No. 3.35, 11
work page 2024
-
[3]
Igor Balla, Minimum density of union-closed families, 2011, arXiv:1106.0369
work page Pith review arXiv 2011
-
[4]
Igor Balla, B\' e la Bollob\' a s, and Tom Eccles, Union-closed families of sets, J. Combin. Theory Ser. A 120 (2013), no. 3, 531--544
work page 2013
-
[5]
Henning Bruhn, Pierre Charbit, Oliver Schaudt, and Jan Arne Telle, The graph formulation of the union-closed sets conjecture, European J. Combin. 43 (2015), 210--219
work page 2015
-
[6]
Henning Bruhn and Oliver Schaudt, The journey of the union-closed sets conjecture, Graphs Combin. 31 (2015), no. 6, 2043--2074
work page 2015
-
[7]
Stijn Cambie, Better bounds for the union-closed sets conjecture using the entropy approach, 2022, arXiv:2212.12500
arXiv 2022
-
[8]
Thomas M Cover, Elements of information theory, John Wiley & Sons, 1999
work page 1999
Show all 22 references
-
[9]
Justin Gilmer, A constant lower bound for the union-closed sets conjecture, 2022, arXiv:2211.09055
2022 arXiv
-
[10]
Emanuel Knill, Graph generated union-closed families of sets, 1994, arXiv:math/9409215
1994 arXiv
-
[11]
Jingbo Liu, Improving the lower bound for the union-closed sets conjecture via conditionally iid coupling, 2024 58th Annual Conference on Information Sciences and Systems (CISS) (2023), 1--6
2023
-
[12]
Robert Morris, F C -families and improved bounds for F rankl's conjecture , European J. Combin. 27 (2006), no. 2, 269--282
2006
-
[13]
Nicolas Nagel, Notes on the union closed sets conjecture, 2023, arXiv:2208.03803
2023 arXiv
-
[14]
Luke Pebody, Extension of a method of G ilmer , 2022, arXiv:2211.13139
2022 arXiv
-
[15]
PolyMath, Frankl's union-closed conjecture, https://www.michaelnielsen.org/polymath/index.php?title=Frankl
-
[16]
David Reimer, An average set size theorem, Combin. Probab. Comput. 12 (2003), no. 1, 89--93
2003
-
[17]
16 (2000), no
J\" u rgen Reinhold, Frankl's conjecture is true for lower semimodular lattices, Graphs Combin. 16 (2000), no. 1, 115--116
2000
-
[18]
Ian Roberts and Jamie Simpson, A note on the union-closed sets conjecture, Australas. J. Combin. 47 (2010), 265--267
2010
-
[19]
Will Sawin, An improved lower bound for the union-closed set conjecture, 2023, arXiv:2211.11504
2023 arXiv
-
[20]
Vaughan, Families implying the F rankl conjecture , European J
Theresa P. Vaughan, Families implying the F rankl conjecture , European J. Combin. 23 (2002), no. 7, 851--860
2002
-
[21]
Bojan Vu c kovi\' c and Miodrag Z ivkovi\' c , The 12-element case of F rankl's C onjecture , IPSI BgD Transactions on Internet Research 13 (2017), 65--71
2017
-
[22]
199 (1999), no
Piotr W \' o jcik, Union-closed families of sets, Discrete Math. 199 (1999), no. 1-3, 173--182
1999
Reviewed August 11, 2026 · model on record in the stance chip above.
Discussion (0). Continue with ORCID to comment.