Optimal two-element frequency constants for finite union-closed families (Question 1 of Hu–Shi–Zhou)

A finite family of sets is union-closed if it contains the union of any two of its members. Hu, Shi and Zhou (arXiv:2507.11008) asked whether there exist absolute constants $c_1\ge c_2>0$, independent of the family, such that every finite union-closed family whose ground set has at least two elements contains two distinct elements lying in at least $c_1$ and $c_2$ fractions of its members, and conjectured that the best possible pair is $(1/2,1/3)$. The existence of such constants is settled, and the second coordinate is now a theorem — Das and Wu proved Nagel's conjecture for all $k\ge2$, so the second-most-frequent element always lies in at least a third of the members, with equality attained by the three-set family $\{\emptyset,\{1\},\{1,2\}\}$. The open part of Question 1 is the first coordinate: whether $c_1$ can equal $1/2$, which is equivalent to Frankl's union-closed sets conjecture, a central open problem of extremal set theory. The best rigorous constant to date is approximately $0.382346$, obtained by entropy methods that plateau well below $1/2$. The stronger $S_2$-conjecture of Cui and Hu — two elements each in at least half the sets whenever the smallest nonempty member has size at least two — also remains open.

Background

A family $\mathscr{F}$ of finite sets is called union-closed if $A\cup B\in\mathscr{F}$ whenever $A,B\in\mathscr{F}$. For an element $x$ of the ground set $\cup_{A\in\mathscr{F}}A$, its frequency is the fraction $|\mathscr{F}_x|/|\mathscr{F}|$, where $\mathscr{F}_x=\{A\in\mathscr{F}:x\in A\}$ is the subfamily of members containing $x$. Frankl's union-closed sets conjecture (posed by Péter Frankl in 1979) asserts that every finite union-closed family other than the trivial family $\{\emptyset\}$ has an element of frequency at least $1/2$; it remains open despite a large literature surveyed by Bruhn and Schaudt, with verified special cases including families whose ground set has at most 12 elements (Poonen proved the cases of at most 7 ground elements or at most 28 sets, later sharpened by others). Two quantitative refinements frame the present problem.

First, the approximate version. Gilmer proved in 2022, by an information-theoretic (entropy) argument, that some element always has frequency at least a fixed absolute constant — the first such bound; his paper stated $0.01$, and the same argument was sharpened by Sawin and, independently, by Chase and Lovett to $(3-\sqrt5)/2\approx0.38197$, the ceiling of the basic version of the method. Cambie improved the rigorous constant to about $0.382346$ and showed that the coupling refinement suggested by Sawin caps at essentially this value (a coupling being a chosen joint distribution for the two random sets sampled inside the entropy argument); Liu obtained about $0.38271$ conditional on numerically verified hypotheses. No current technique approaches $1/2$.

Second, the $k$-th frequency version. Nagel conjectured that if the ground set has at least $k$ elements, then the $k$-th most frequent element has frequency at least $1/(2^{k-1}+1)$; the case $k=1$ is Frankl's conjecture. Das and Wu proved this for all $k\ge2$: the $k$-th most frequent element lies in at least a $1/(2^{k-1}+1)$ fraction of the members, with equality only for the near-$k$-cubes $\mathscr{F}_k=2^{[k-1]}\cup\{S\}$ (all subsets of a fixed $(k-1)$-element set, together with one further set $S$ strictly containing that set). For $k=2$ this says the second-most-frequent element always has frequency at least $1/3$, and the near-2-cube $\{\emptyset,\{1\},\{1,2\}\}$ shows $1/3$ cannot be improved. On the other side, in the full cube $2^{[t]}$ (all subsets of a $t$-element set) every element has frequency exactly $1/2$, so no universal first coordinate can exceed $1/2$.

Within this landscape, Hu, Shi and Zhou posed their Question 1, asking for the best pair of constants in a two-element guarantee (stated below). Their Lemma 1.1 transfers frequency lower bounds through the deletion map $A\mapsto A\setminus\{i\}$, which already yields unconditional pairs such as $(0.38234,0.23635)$. They also recalled a stronger conjecture of Cui and Hu, the $S_2$-version: writing $T(\mathscr{F})$ for the size of the smallest nonempty member of $\mathscr{F}$, if $T(\mathscr{F})\ge2$ (no singleton members), then at least two elements have frequency at least $1/2$. Kabela, Polák and Teska showed that this two-element strengthening stands between Frankl's conjecture and Poonen's conjecture (which predicts that if a family has a unique abundant element — an element contained in more than half the members — then that element lies in every member), proved it when the smallest member has size at least $n-3$ with $n$ the largest member size, and constructed families with only $k-1$ abundant elements for $3\le k\le n-4$ and $n\ge9$, delimiting possible strengthenings.

Problem Statement

Let $\mathscr{F}$ be a finite union-closed family of sets with ground set $\cup_{A\in\mathscr{F}}A=\{1,2,\dots,m\}$ and $m\ge2$, and for each element $x$ write $\mathscr{F}_x=\{A\in\mathscr{F}:x\in A\}$. Question 1 of Hu–Shi–Zhou asks: do there exist absolute constants $c_1\ge c_2>0$, independent of $m$ and of $\mathscr{F}$, such that $\mathscr{F}$ always contains two distinct elements $i,j$ with $|\mathscr{F}_i|/|\mathscr{F}|\ge c_1$ and $|\mathscr{F}_j|/|\mathscr{F}|\ge c_2$; and if so, what is the coordinatewise maximal (best possible) pair $(c_1^*,c_2^*)$? The authors conjecture the optimal pair is $(1/2,1/3)$.

The existence clause is settled affirmatively, and $c_2^*=1/3$ is now a theorem (Das–Wu's resolution of Nagel's conjecture for $k=2$); both coordinates are individually capped — the full cube shows $c_1^*\le1/2$ and the near-2-cube $\{\emptyset,\{1\},\{1,2\}\}$ shows $c_2^*\le1/3$. The open part of Question 1 is the first coordinate: determine $c_1^*$ exactly. In particular, prove or disprove that every finite union-closed family whose ground set has at least two elements contains an element of frequency at least $1/2$ together with a second, distinct element of frequency at least $1/3$ — a statement now equivalent to Frankl's union-closed sets conjecture. If the answer is negative, the task is to determine $c_1^*$, the infimum over all such families of the maximum element frequency, exactly. Improving the current rigorous constant $c_1\approx0.382346$ is partial progress toward the first coordinate but does not by itself resolve the task, which requires determining $c_1^*$ exactly; the stronger $S_2$-version of Cui and Hu (two elements each of frequency $\ge1/2$ whenever the smallest nonempty member has size $\ge2$) is a strictly stronger open conjecture that implies Frankl's conjecture.

References

1. Ze-Chun Hu, Yi-Ding Shi, Qian-Qian Zhou, "A lemma on a finite union-closed family of finite sets and its applications", arXiv:2507.11008 (2025), https://arxiv.org/abs/2507.11008
2. Shagnik Das, Saintan Wu, "Frequent elements in union-closed set families", arXiv:2412.03862 (2024; v3 July 2025), https://arxiv.org/abs/2412.03862
3. Saintan Wu, "Further analysis on the second frequency of union-closed set families", arXiv:2412.03863 (2024), https://arxiv.org/abs/2412.03863
4. Justin Gilmer, "A constant lower bound for the union-closed sets conjecture", arXiv:2211.09055 (2022), https://arxiv.org/abs/2211.09055
5. Will Sawin, "An improved lower bound for the union-closed set conjecture", arXiv:2211.11504 (2022), https://arxiv.org/abs/2211.11504
6. Zachary Chase, Shachar Lovett, "Approximate union closed conjecture", arXiv:2211.11689 (2022), https://arxiv.org/abs/2211.11689
7. Stijn Cambie, "Better bounds for the union-closed sets conjecture using the entropy approach", arXiv:2212.12500 (2022), https://arxiv.org/abs/2212.12500
8. Jingbo Liu, "Improving the Lower Bound for the Union-closed Sets Conjecture via Conditionally IID Coupling", arXiv:2306.08824 (2023), https://arxiv.org/abs/2306.08824
9. Nicolas Nagel, "Notes on the union closed sets conjecture", arXiv:2208.03803 (2022), https://arxiv.org/abs/2208.03803
10. Adam Kabela, Michal Polák, Jakub Teska, "The number of abundant elements in union-closed families without small sets", arXiv:2212.09279 (2022), https://arxiv.org/abs/2212.09279
11. Zhen Cui, Zechun Hu, "Two stronger versions of the union-closed sets conjecture", Advances in Mathematics (Beijing) 50(6), 829-851 (2021), DOI 10.11845/sxjz.2020068b, https://doi.org/10.11845/sxjz.2020068b (Zbl 1513.05030)
12. Bjorn Poonen, "Union-closed families", Journal of Combinatorial Theory, Series A 59(2), 253-268 (1992), DOI 10.1016/0097-3165(92)90068-6, https://doi.org/10.1016/0097-3165(92)90068-6
13. Henning Bruhn, Oliver Schaudt, "The Journey of the Union-Closed Sets Conjecture", Graphs and Combinatorics 31(6), 2043-2074 (2015), DOI 10.1007/s00373-014-1515-0, https://doi.org/10.1007/s00373-014-1515-0