The UC_x ladder for Frankl's union-closed conjecture: Bouchard's subfamily-existence Question 2.2 and implication Question 2.5

Frankl's union-closed sets conjecture — every finite union-closed family of sets has an element contained in at least half of its member sets — is equivalent to the extremal rung $x=n$ of a ladder of statements $\mathrm{UC}_x$ introduced by Bouchard: for a union-closed family with ground set $[n]$, $\mathrm{UC}_x$ asks for a subset $B$ of the ground set of size $n-x+1$ such that at least as many member sets contain all of $B$ as are disjoint from $B$. Only the rungs with $x\le\lceil n/3\rceil+1$ are proven. This problem asks for a resolution of the two questions that control the rest of the ladder: Question 2.2, whether every finite family $\mathcal{F}$ with $|\mathcal{F}|>|\bigcup\mathcal{F}|+1$ (with no union-closure assumption) has a proper subfamily $\mathcal{F}'$ with $|\mathcal{F}|-|\mathcal{F}'|=|\bigcup\mathcal{F}'|+1$, an affirmative answer to which would extend the proven rungs to $x\le\lfloor n/2\rfloor+1$; and Question 2.5, whether, for every union-closed family, the truth of $\mathrm{UC}_n$ for that family forces $\mathrm{UC}_x$ for all $x\in[n-1]$, so that a proof of Frankl's conjecture would settle the whole ladder. Both questions remain open.

Background

A family of sets $\mathcal{A}$ is union-closed if it is finite and nonempty, its member sets are finite and distinct (at least one of them nonempty), and the union of any two member sets is again a member set. Frankl's union-closed sets conjecture (1979), a central open problem of extremal combinatorics, asserts that every union-closed family contains an element belonging to at least half of its member sets. The best general results are constant lower bounds on the maximum element frequency: Gilmer showed by an entropy method that some element appears in at least $0.01\,|\mathcal{A}|$ member sets, and Sawin improved the constant to $(3-\sqrt{5})/2\approx 0.38$. These are results about single-element frequencies and do not settle the conjecture.

Bouchard (arXiv:2310.02482) organizes the conjecture into a ladder of statements. Write $[n]=\bigcup_{A\in\mathcal{A}}A$ for the ground set of the family (an $n$-element set); for a subset $B\subseteq[n]$ let $\mathcal{A}_B=\{A\in\mathcal{A}: A\cap B=B\}$ denote the member sets containing all of $B$, and $\mathcal{A}_{\underline{B}}=\{A\in\mathcal{A}: A\cap B=\emptyset\}$ those disjoint from $B$. For $x\in[n]$, the statement $\mathrm{UC}_x$ reads: there exists a subset $B\in\binom{[n]}{n-x+1}$ (a set of $n-x+1$ ground elements) with $|\mathcal{A}_B|\ge|\mathcal{A}_{\underline{B}}|$. The rung $x=n$ asks this for a singleton $B=\{b\}$, which is exactly Frankl's conjecture, since $|\mathcal{A}_{\{b\}}|\ge|\mathcal{A}_{\underline{\{b\}}}|$ says that element $b$ lies in at least half the member sets. The rung $x=1$, at the other end, holds trivially for every union-closed family: the union of all member sets is itself a member set, so exactly one member contains all of $[n]$, while at most one member — the empty set, if it belongs to the family — is disjoint from $[n]$. Implications between rungs are known only in limited ranges, and this is where the open questions live.

Bouchard proves $\mathrm{UC}_x$ for all $x\le\lceil n/3\rceil+1$ (Theorem 2.1) by a constructive induction that climbs from the trivial rung $x=1$ and whose guiding quantity $n-x-|Z|$ hits $0$ exactly at this boundary, so the method stops there; the paper also proves the adjacent implication $\mathrm{UC}_{n-1}\Rightarrow\mathrm{UC}_n$ for every family (Theorem 2.4). Two further questions control progress. Question 2.2 is a purely additive existence statement about arbitrary finite families with no union-closure hypothesis: if $|\mathcal{F}|>|\bigcup_{F\in\mathcal{F}}F|+1$, must there exist a proper subfamily $\mathcal{F}'\subsetneq\mathcal{F}$ with $|\mathcal{F}|-|\mathcal{F}'|=|\bigcup_{F\in\mathcal{F}'}F|+1$? Proposition 2.3 of the paper shows that an affirmative answer would push the proven range of the ladder up to $x\le\lfloor n/2\rfloor+1$. Question 2.5 asks whether the Frankl rung implies the others, family by family: for a union-closed family $\mathcal{A}$ with $n>1$ that satisfies $\mathrm{UC}_n$, must $\mathcal{A}$ also satisfy $\mathrm{UC}_x$ for every $x\in[n-1]$? An affirmative answer would mean that a proof of Frankl's conjecture settles every rung of the ladder at once; the adjacent case $\mathrm{UC}_n\Rightarrow\mathrm{UC}_{n-1}$, the converse of Theorem 2.4, remains open. Recent adjacent work stays on different axes — Das and Wu confirmed a 2022 conjecture of Nagel that the $k$-th most frequent element of a union-closed family lies in at least $|\mathcal{A}|/(2^{k-1}+1)$ member sets, and Tian proved Frankl's conjecture for families of height at most four in the empty-set-free formulation (height being the length of the longest chain of member sets under inclusion) — and none of it addresses these two ladder questions.

Problem Statement

Resolve the following two questions from Bouchard (arXiv:2310.02482), which together determine how far the $\mathrm{UC}_x$ ladder extends beyond the currently proven range $x\le\lceil n/3\rceil+1$ and how its rungs relate to Frankl's conjecture.

1. Question 2.2 (subfamily existence). Let $\mathcal{F}$ be a finite family of finite sets — not necessarily union-closed, and with members not necessarily distinct, so that $|\mathcal{F}|$ counts repeated members with multiplicity — satisfying $|\mathcal{F}|>|\bigcup_{F\in\mathcal{F}}F|+1$. Must there exist a proper subfamily $\mathcal{F}'\subsetneq\mathcal{F}$ such that $|\mathcal{F}|-|\mathcal{F}'|=|\bigcup_{F\in\mathcal{F}'}F|+1$? An affirmative answer, combined with Proposition 2.3 of the source paper, establishes $\mathrm{UC}_x$ for every union-closed family with $x\le\lfloor n/2\rfloor+1$; a negative answer is witnessed by a single explicit finite family satisfying the premise for which no proper subfamily meets the equality.

2. Question 2.5 (implication among rungs). For every union-closed family $\mathcal{A}$ with ground set $[n]$ and $n>1$: if $\mathrm{UC}_n$ holds for $\mathcal{A}$ — that is, if some element of $[n]$ lies in at least half of the member sets of $\mathcal{A}$ — must $\mathrm{UC}_x$ also hold for $\mathcal{A}$ for every $x\in\{1,\dots,n-1\}$, i.e. must there exist, for each such $x$, a subset $B\subseteq[n]$ of size $n-x+1$ with $|\mathcal{A}_B|\ge|\mathcal{A}_{\underline{B}}|$? An affirmative answer would show that Frankl's conjecture, if true, implies every rung of the ladder (family by family), so that any proof of Frankl's conjecture would settle the entire ladder; a negative answer is witnessed by a single explicit union-closed family that satisfies $\mathrm{UC}_n$ but violates $\mathrm{UC}_x$ for some $x\in\{1,\dots,n-1\}$. (The source paper proves the adjacent implication $\mathrm{UC}_{n-1}\Rightarrow\mathrm{UC}_n$ for every family; these converse directions remain open.)

A complete answer resolves both questions: each either by a proof valid for all qualifying families, or by an explicit finite counterexample family — for Question 2.2, a family satisfying $|\mathcal{F}|>|\bigcup\mathcal{F}|+1$ for which no proper subfamily achieves the equality; for Question 2.5, a union-closed family satisfying $\mathrm{UC}_n$ but violating some $\mathrm{UC}_x$. The common objective is a complete determination of the reach and the implication structure of the $\mathrm{UC}_x$ ladder below its top rung; neither question, once resolved, settles Frankl's conjecture itself.

References

1. Christopher Bouchard, "Conjectures on union-closed families of sets", arXiv:2310.02482 (2023), https://arxiv.org/abs/2310.02482 (published as Graduate Journal of Mathematics, Volume 9, Issue 1 (2024), 51–59)
2. Justin Gilmer, "A constant lower bound for the union-closed sets conjecture", arXiv:2211.09055 (2022), https://arxiv.org/abs/2211.09055
3. Will Sawin, "An improved lower bound for the union-closed set conjecture", arXiv:2211.11504 (2022), https://arxiv.org/abs/2211.11504
4. Shagnik Das and Saintan Wu, "Frequent elements in union-closed set families", arXiv:2412.03862 (2024), https://arxiv.org/abs/2412.03862
5. Christopher Bouchard, "An averaging result for union-closed families of sets", arXiv:2509.12537 (2025), https://arxiv.org/abs/2509.12537
6. Chenxiao Tian, "Frankl's Conjecture at Height Four and the Structure of Height-Five Counterexamples", arXiv:2608.25147 (2026), https://arxiv.org/abs/2608.25147
7. Ze-Chun Hu, Yi-Ding Shi, and 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
8. Roberto Demontis, "The union-closed set conjecture is true", arXiv:2405.03731 (2024), https://arxiv.org/abs/2405.03731 (also published as DOI 10.33140/crsm.03.02.04)