Frankl's Union-Closed Sets Conjecture

Frankl's Union-Closed Sets Conjecture

Definitions and mathematical statement

A finite family $\mathcal F$ of finite sets is union-closed if $A\cup B\in\mathcal F$ whenever $A,B\in\mathcal F$. The standard nondegeneracy condition is $\bigcup\mathcal F\ne\varnothing$. Frankl's conjecture then asserts \[ \exists x\in\bigcup\mathcal F:\qquad \#\{A\in\mathcal F:x\in A\}\ge\frac{|\mathcal F|}{2}. \] The empty set may itself be a member. Scope qualification: the catalogue says "other than the empty set," which is ambiguous for a family. Excluding only the empty family is insufficient, because $\mathcal F=\{\varnothing\}$ is union-closed and has no element at all. The displayed statement explicitly excludes both degenerate cases, as required for the usual conjecture. The family is a set of sets, not a multiset with arbitrary weights.

Formulation and concept references: [S1].

Short English statement

In every nondegenerate finite union-closed family, must some element occur in at least half of the sets?

Research attempt

Scope, process, and first gap.

This GPT-6 Astra Ultra attempt, prepared on 27 September 2026, uses injections induced by union, reduction to a separating family, and explicit frequency counts. The family is unweighted and its union is nonempty. The partial arguments are classical and elementary; the separating-family mechanism is attributed to the work discussed in [S1] and to Falgas--Ravry [E1]. They are reconstructed rather than claimed as new.

The survey [S1] predates the later entropy-based positive constant lower bounds, including Gilmer's [E2]. Thus its historical discussion of a constant-fraction weakening cannot be read as current evidence that even that weakening is open. A positive constant fraction is still different from the exact one-half conclusion sought here. Our first gap is obtaining one element of frequency at least $|\mathcal F|/2$ when neither a small member nor the separating-family size criterion proved below applies.

Singletons give an injection proof.

Let $m=|\mathcal F|$ and suppose $\{x\}\in\mathcal F$. Map every member not containing $x$ to $A\cup\{x\}$. Union-closure puts the image in $\mathcal F$, it contains $x$, and deleting $x$ recovers $A$. The map is injective. Therefore \[ |\{A:x\notin A\}|\le|\{A:x\in A\}|, \] which is exactly the conjectured bound for $x$. This includes the case that the empty set is a member.

More generally, if a fixed nonempty $C\in\mathcal F$ has size $k$, the map $A\mapsto A\cup C$ has fibers of size at most $2^k$: preimages have the same trace outside $C$, with at most $2^k$ possible traces inside it. Its image consists of members containing all of $C$. Thus every $x\in C$ occurs in at least $m/2^k$ members. This is valid but becomes weak when $k$ grows. Treating the map as injective for general $C$ would discard precisely these collisions.

A two-element member also suffices.

Suppose $\{a,b\}\in\mathcal F$, and let $n_{00},n_{10},n_{01},n_{11}$ count members according to membership of $a,b$. The map \[ A\longmapsto A\cup\{a,b\} \] is injective on the $00$ class into the $11$ class, because their traces outside $\{a,b\}$ distinguish the preimages. Hence $n_{11}\ge n_{00}$. Adding the two frequencies gives \[ d(a)+d(b)=n_{10}+n_{01}+2n_{11} \ge n_{00}+n_{10}+n_{01}+n_{11}=m. \] At least one is therefore at least $m/2$. Notice that this proof does not require either singleton to belong to the family, and does not require the two frequencies to be equal.

A counterexample to the full conjecture, if one exists, consequently has no one-element or two-element member. This is a genuine structural exclusion, but it leaves families whose smallest nonempty member is larger.

Removing indistinguishable elements.

Two elements of the union are twins if every member contains both or neither. If $x,y$ are twins, delete $y$ from every set. The resulting family remains union-closed, and the deletion map is injective on members: two preimages with the same image have the same membership of $x$, which forces the same membership of $y$.

Thus the number of members and the frequencies of the retained elements are unchanged. Repeating gives a separating family, meaning that each two distinct remaining elements are distinguished by some member. At least one element remains, because the original union was nonempty. A frequent retained element is equally frequent in the original family. This reduction changes the ground-set size but not the frequency denominator; counting deleted twins as independent witnesses would be incorrect.

A complete small-family criterion.

Now suppose $\mathcal F$ separates its $n$ ground elements, ordered so that \[ d(x_1)\le d(x_2)\le\cdots\le d(x_n). \] For $i<j$, there exists a member containing $x_j$ but not $x_i$. Otherwise every member containing $x_j$ would contain $x_i$, giving $d(x_j)\le d(x_i)$. Together with the ordering this forces equal occurrence sets, contradicting separation.

For each $i<n$, take the union of chosen witnesses containing $x_j$ but not $x_i$, over all $j>i$, and call it $A_i$. Then \[ A_i\in\mathcal F,\qquad x_i\notin A_i,\qquad \{x_{i+1},\ldots,x_n\}\subseteq A_i. \] These $n-1$ members are distinct: if $i<j$, then $x_j\in A_i$ but $x_j\notin A_j$. The full union $U=\bigcup\mathcal F$ is itself in $\mathcal F$ by finite union-closure and differs from every $A_i$. All these $n$ members contain $x_n$. Therefore \[ d(x_n)\ge n. \tag{411.1} \] For $n=1$, the same conclusion follows from the member $U$, without constructing any $A_i$.

Consequently every separating union-closed family with $m\le2n$ satisfies the conjecture. For an arbitrary family the criterion applies with the number of twin classes after reduction. It is not valid to keep the original, possibly inflated number of ground elements after identifying twins.

The witness construction also proves \[ \sum_{A\in\mathcal F}|A| =\sum_x d(x)\ge n+(n-1)+\cdots+1 =\frac{n(n+1)}2. \] This is additional incidence information, but it is an absolute bound in terms of $n$. When $m$ is much larger than $n$, it does not force an individual frequency to reach $m/2$.

Why a direct extension stalls.

The witnesses $A_i$ are indexed by elements, not by all members. The construction produces $n$ sets containing $x_n$, and supplies no injection from every set avoiding $x_n$ into that list. Additional witnesses may coincide with old ones. Thus the numerical implication from $411.1$ to the half bound uses $m\le2n$ essentially.

Trying to average over unions has a different issue. If $A,B$ are independent uniform members, then $A\cup B$ lies in the family, but is not generally a uniform member. An element of frequency proportion $p$ belongs to the union with probability $2p-p^2$, which is larger than $p$ for $0<p<1$. That increase describes a changed distribution, not a contradiction. Even for the full power set, uniform membership probabilities are $1/2$, while those of the union are $3/4$. Closure controls the support of the union distribution, not its uniformity.

Why arbitrary weights change the assertion.

Consider the union-closed family \[ \{\varnothing,\{a\},\{a,b\}\}. \] Unweighted, $a$ occurs in two of its three members. Give these three sets weights $9,1,1$, respectively. Then the weighted occurrence proportions are $2/11$ and $1/11$, both below one-half. This is not a counterexample to Frankl's conjecture: arbitrary weights are excluded by the statement. It shows that replacing cardinalities by a convenient nonuniform measure requires a separate theorem and cannot be assumed during an entropy or averaging argument.

Verification and outcome.

The exact checks enumerate all set families on ground sets of size at most four, retain the union-closed nondegenerate ones, and verify the injection, two-element-member, twin-reduction, and separating-witness conclusions. They do not extrapolate small-ground-set truth to all families.

UNRESOLVED. The conjecture is proved here for families containing a member of size one or two, and for separating families with at most twice as many members as ground elements. A possible counterexample must survive these reductions. The remaining gap is an argument yielding the half-frequency bound outside these regimes without changing the uniform measure on the family.

Sources

• S1 Bruhn, H., Schaudt, O., 'The journey of the union-closed sets conjecture', arXiv:1309.3297 $2013$. https://arxiv.org/pdf/1309.3297. Formulation source.
• E1 V. Falgas--Ravry, Minimal Weight in Union-Closed Families, Electronic Journal of Combinatorics 18 $2011$, P95. https://arxiv.org/abs/1101.2589.
• E2 J. Gilmer, A constant lower bound for the union-closed sets conjecture, arXiv:2211.09055. https://arxiv.org/abs/2211.09055. Later constant-fraction progress, distinct from the one-half target; consulted 27 September 2026.

Partial result on TOP-55

0. Statement and conventions

The problem statement is kept unchanged. A finite family $\mathcal F$ of finite sets is *union-closed* if $A\cup B\in\mathcal F$ for all $A,B\in\mathcal F$. It is *nondegenerate* if $U(\mathcal F):=\bigcup\mathcal F\neq\varnothing$. Frankl's conjecture asserts that for every nondegenerate union-closed $\mathcal F$,
$$\exists x\in U(\mathcal F):\qquad d(x):=\#\{A\in\mathcal F: x\in A\}\ \ge\ \tfrac12|\mathcal F|.$$
The claims below cover only the stated special cases. They are not a proof of the conjecture.

Notation:
- $m=|\mathcal F|$ and $D(\mathcal F)=\max_x d(x)$.
- Two elements $x,y\in U(\mathcal F)$ are *twins* if they lie in exactly the same members. $n=n(\mathcal F)$ is the number of twin classes.
- $\mathcal F$ is *separating* if it has no twins, i.e. $n=|U(\mathcal F)|$.
- $y$ *dominates* $x$ if every member containing $x$ also contains $y$.
- $\psi:=\frac{3-\sqrt5}{2}\approx0.381966$. Note that $\frac{2}{1-\psi}=1+\sqrt5$.

**Input (E)** [AHS, Theorem 2 with Claim 4; also CL, Sa]. Every finite union-closed family $\mathcal G$ with at least one nonempty member has an element contained in at least $\psi|\mathcal G|$ members of $\mathcal G$.

1. Results

**Theorem 1.** Let $\mathcal F$ be a nondegenerate finite union-closed family with $m$ members and $n$ twin classes. If $m\le(1+\sqrt5)(n-1)$, then some element lies in at least $m/2$ members of $\mathcal F$.

**Theorem 2.** Let $\gamma\in(0,1]$ be such that every finite union-closed family $\mathcal G$ with a nonempty member has an element in at least $\gamma|\mathcal G|$ members. Then every nondegenerate finite union-closed $\mathcal F$ with $m$ members and $n$ twin classes satisfies
$$D(\mathcal F)\ \ge\ \min\Big\{\,m-1,\ \frac{\gamma m+n-1}{1+\gamma}\Big\}.$$
Consequently, if $m\le \frac{2}{1-\gamma}(n-1)$ then $D(\mathcal F)\ge m/2$.

**Corollary 3 (sparse families beat the entropy constant).** For every nondegenerate union-closed $\mathcal F$ with $m\ge2$:
$$\frac{D(\mathcal F)}{m}\ \ge\ \min\Big\{1-\tfrac1m,\ \frac{\psi+(n-1)/m}{1+\psi}\Big\}.$$
The right-hand side is $>\psi$ whenever $m<\frac{n-1}{\psi^2}=\frac{7+3\sqrt5}{2}(n-1)\approx6.854(n-1)$. For example, $m\le4(n-1)$ gives $D\ge\frac{\psi+1/4}{1+\psi}m\approx0.457\,m$.

**Corollary 4 (lattice form).** Let $L$ be a finite lattice with $|L|\ge2$, and let $M(L)$ be its set of meet-irreducible elements (the top excluded). If $|L|\le(1+\sqrt5)(|M(L)|-1)$, then some $\mu\in M(L)$ has $|{\downarrow}\mu|\le|L|/2$. Dually, if $|L|\le(1+\sqrt5)(|J(L)|-1)$, then some join-irreducible $j$ has $|{\uparrow}j|\le|L|/2$.

2. Proofs

**Reduction to separating families.** Let $x,y$ be twins. Deleting $y$ from every member is injective on $\mathcal F$: two members with the same image have the same membership of $x$, hence of $y$. The deletion also preserves union-closure, $m$, and the frequencies of all remaining elements. Repeating gives a separating family on $n$ elements with the same $m$ and the same $D$. So we may assume $\mathcal F$ is separating with $U(\mathcal F)=\{x_1,\dots,x_n\}$, labelled so that $d(x_1)\le\dots\le d(x_n)$. Then $D=d(x_n)$.

**Step 1 (Falgas-Ravry [FR, Lemmas 1–2]).** For $1\le i<j\le n$, some member contains $x_j$ but not $x_i$.

*Proof.* Otherwise $\{A: x_j\in A\}\subseteq\{A:x_i\in A\}$, so $d(x_j)\le d(x_i)\le d(x_j)$. Then the two occurrence sets are equal, which contradicts separation. Ties in the labelling are harmless.

Define the following sets:
- $A_0:=U(\mathcal F)$. It belongs to $\mathcal F$ as the union of all members.
- For $1\le i\le n-1$, $A_i:=\bigcup\{A\in\mathcal F: x_i\notin A\}$. The union is over a nonempty collection by Step 1 with $j=n$, so $A_i\in\mathcal F$.

These sets have three properties:
- (P1) $x_i\notin A_i$ for $i\ge1$.
- (P2) $\{x_{i+1},\dots,x_n\}\subseteq A_i$ for $0\le i\le n-1$. For $i\ge1$, each witness from Step 1 is one of the sets in the union.
- (P3) $A_0,\dots,A_{n-1}$ are distinct. For $i<j$ we have $x_j\in A_i$ and $x_j\notin A_j$.

So $\mathcal S:=\{A_0,\dots,A_{n-1}\}$ consists of $n$ distinct members, each containing $x_n$.

**Step 2 (Y. Hu's domination lemma [Hu, Lemma 1]).** For $1\le i\le n-1$ let $s(i):=\#\{0\le j\le n-1: x_i\in A_j\}$. Then $s(i)\le n-1$. Moreover, every $x_i$ with $1\le i\le n-1$ is dominated by some $x_k$ with $1\le k\le n-1$ and $s(k)=n-1$.

*Proof.* By (P2), $x_i\in A_j$ for every $j<i$, including $j=0$. By (P1), $x_i\notin A_i$. So $s(i)\le n-1$, with equality unless $x_i\notin A_j$ for some $j$ with $i<j\le n-1$.

Suppose $x_i\notin A_j$ for such a $j$. Then no member contains $x_i$ without $x_j$, since such a member would lie inside $A_j$. So $x_j$ dominates $x_i$. If $s(j)<n-1$, repeat the argument with $j$. The index strictly increases and stays $\le n-1$, so the process stops. Note that $s(n-1)=n-1$ by the first sentence of this proof. Since domination is transitive, the final $x_k$ dominates $x_i$.

**Step 3 (proof of Theorem 2).** Let $\mathcal B:=\{A\in\mathcal F: x_n\notin A\}$. It is union-closed, $|\mathcal B|=m-D$, and $\mathcal B\cap\mathcal S=\varnothing$ because every set in $\mathcal S$ contains $x_n$.

*Case (i): $\mathcal B$ has no nonempty member.* Then $|\mathcal B|\le1$, so $D\ge m-1$.

*Case (ii): $\mathcal B$ has a nonempty member.* Let $\beta:=\max_x\#\{A\in\mathcal B:x\in A\}\ge1$. It is attained at some $x_i$ with $i\le n-1$, because $x_n$ lies in no member of $\mathcal B$. Take $x_k$ as in Step 2 (if $s(i)=n-1$, take $k=i$).

Since $x_k$ dominates $x_i$, we get $\#\{A\in\mathcal B: x_k\in A\}\ge\beta$. Because $\mathcal B$ and $\mathcal S$ are disjoint subfamilies of $\mathcal F$,
$$D\ \ge\ d(x_k)\ \ge\ \#\{A\in\mathcal B:x_k\in A\}+s(k)\ \ge\ \beta+n-1.$$
The hypothesis on $\gamma$, applied to $\mathcal B$, gives $\beta\ge\gamma|\mathcal B|=\gamma(m-D)$. Hence $(1+\gamma)D\ge\gamma m+n-1$.

*Consequence.* Suppose $m\le\frac{2}{1-\gamma}(n-1)$. Then $n\ge2$, and $m\ge n\ge2$ by (P3), so in Case (i) $D\ge m-1\ge m/2$. In Case (ii), $n-1\ge\frac{(1-\gamma)m}{2}$, hence
$$D\ \ge\ \frac{\gamma m+\frac{(1-\gamma)m}{2}}{1+\gamma}=\frac m2.$$
This proves Theorem 2. $\square$

**Proof of Theorem 1.** Apply Theorem 2 with $\gamma=\psi$, which is valid by (E); note that $\frac{2}{1-\psi}=1+\sqrt5$. $\square$

**Proof of Corollary 3.** Divide Theorem 2 (with $\gamma=\psi$) by $m$. Then $1-\frac1m\ge\frac12>\psi$ for $m\ge2$, and $\frac{\psi+t}{1+\psi}>\psi\iff t>\psi^2$. $\square$

**Proof of Corollary 4.** For $a\in L$ put $A(a):=\{\mu\in M(L): a\not\le\mu\}$, and let $\mathcal F_L:=\{A(a):a\in L\}$.
- $A(a\vee b)=A(a)\cup A(b)$, so $\mathcal F_L$ is union-closed.
- $a=\bigwedge\{\mu\in M(L):a\le\mu\}$, so $a\mapsto A(a)$ is injective and $|\mathcal F_L|=|L|$.
- $U(\mathcal F_L)=A(\hat1)=M(L)\neq\varnothing$, since $|L|\ge2$.
- $\mathcal F_L$ is separating. If $\mu\neq\nu$, we may assume $\nu\not\le\mu$; then $A(\nu)$ contains $\mu$ but not $\nu$.
- The frequency of $\mu$ is $|L|-|{\downarrow}\mu|$.

Theorem 1 with $n=|M(L)|$ gives the claim. The dual statement follows by applying this to the dual lattice. $\square$

3. Comparison and remarks

1. **Previous bounds.** Falgas-Ravry's sets give $D\ge n$, hence the conjecture for separating families with $m\le2n$ (recorded in [BS, Theorem 23] and [Ma, Lemma 1.3]). Maßberg [Ma, Theorem 2.1] proved $m\le2\big(n+\frac{n}{\log_2n-\log_2\log_2n}\big)$; inequality (13) of his proof is slightly stronger. Both are $2n+O(n/\log n)$.
- The new bound $(1+\sqrt5)(n-1)$ exceeds the bound from Maßberg's (13) for every $n\ge34$. Indeed, evaluating his minimum at $k=5$ bounds it by $\frac83n+16$, and $(1+\sqrt5)(n-1)>\frac83n+16$ for $n\ge34$.
- By direct evaluation (my own computation), it also exceeds (13) for $23\le n\le33$, and exceeds his closed form for all $n\ge16$.
- The conjecture is also known for $n\le12$ and for $m\le50$, as stated in [Ma] and [BS] (I did not re-check those sources). Theorem 1 therefore covers families not covered by $m\le50$ or by Maßberg's (13) for every $n\ge27$.
2. **Hu's reduction.** Hu's Theorem 2 [Hu; BS, Theorem 24] says that a factor $c>2$ implies a universal constant $\frac{c-2}{2(c-1)}$. For $c=1+\sqrt5$ this is $\approx0.276<\psi$, so there is no circularity and no new universal constant. Theorem 2 goes the other way, $\gamma\mapsto\frac{2}{1-\gamma}$.
3. **Limit of the method.** Step 3 is Hu's minimal-counterexample argument [Hu, Theorem 1] with the induction hypothesis replaced by (E). With $\gamma=\frac12$ (Frankl's conjecture for $\mathcal B$), Theorem 2 gives $4(n-1)$, consistent with the bound $\approx4n$ for *minimal* counterexamples [Hu, Theorem 1; Roberts–Simpson as cited there]. Any rigorous improvement of the universal constant improves Theorem 1 automatically. Jiang [Ji] claims $0.38288525$ via a computer-assisted proof (not checked here); it would give a factor $\approx3.241$.
4. **Novelty.** On 2026-10-19 I searched arXiv and the web, and read the surveys [BS] and [Ca], the entropy papers [Gi, AHS, CL, Sa], and [Ma, Hu, FR]. I found no statement of Theorem 1 or Theorem 2. The ingredients are all known, so the observation may be folklore.

4. Self-audit

Steps checked:
- ties in the labelling (Step 1);
- that each $A_i$ is well defined and lies in $\mathcal F$;
- distinctness of $\mathcal S$ and $\mathcal S\subseteq\{A: x_n\in A\}$;
- termination of the domination chain, using $s(n-1)=n-1$;
- that the maximiser of $\beta$ is not $x_n$;
- disjointness of $\mathcal B$ and $\mathcal S$;
- the hypotheses of (E) for $\mathcal B$ (finite, union-closed, with a nonempty member; Case (i) handles the rest);
- the trivial cases $n=1$ (Theorem 1 is vacuous) and $m\le2$;
- that the twin reduction preserves $m$ and $D$.

The conclusion is the non-strict $\ge m/2$, as in the statement. The inequalities used are tight only if $\mathcal B$ is extremal for (E), so going beyond the factor $\frac{2}{1-\gamma}$ needs more information about $\mathcal B$ than these steps use.

5. What remains open

- Frankl's conjecture in general.
- Separating families with $m>(1+\sqrt5)(n-1)$.
- Whether the factor can be pushed beyond $\frac{2}{1-\gamma}$, e.g. to $4$, without first improving the universal constant $\gamma$.

References

- [AHS] R. Alweiss, B. Huang, M. Sellke, Improved lower bound for Frankl's union-closed sets conjecture, Electron. J. Combin. 31(3) (2024), P3.35; arXiv:2211.11731.
- [Ca] S. Cambie, Progress on the union-closed conjecture and offsprings in winter 2022–2023, arXiv:2306.12351.
- [BS] H. Bruhn, O. Schaudt, The journey of the union-closed sets conjecture, Graphs Combin. (2015), doi:10.1007/s00373-014-1515-0; arXiv:1309.3297.
- [CL] Z. Chase, S. Lovett, Approximate union closed conjecture, arXiv:2211.11689.
- [FR] V. Falgas-Ravry, Minimal weight in union-closed families, Electron. J. Combin. 18 (2011), #P95.
- [Gi] J. Gilmer, A constant lower bound for the union-closed sets conjecture, arXiv:2211.09055.
- [Hu] Y. Hu, On the union-closed sets conjecture, arXiv:1706.06167.
- [Ji] Y. Jiang, Entropy bounds and global couplings for union-closed families, arXiv:2609.08291.
- [Ma] J. Maßberg, The union-closed sets conjecture for small families, arXiv:1508.05718.
- [Sa] W. Sawin, An improved lower bound for the union-closed set conjecture, arXiv:2211.11504.

---

**Editorial correction (19 October 2026).** In the comparison with Maßberg’s inequality (13), the coverage claim above has been corrected from “every $n\ge23$” to “every $n\ge27$”. The larger real threshold for $23\le n\le26$ does not admit an additional integer family size: the corresponding integer bounds are respectively 71, 74, 77 and 80 under both estimates. The main theorems and the real-threshold comparison are unchanged. The original submission is retained in the source evidence.

Original contributor disclosures

Transcribed from the original public submission. Editor normalization changes only the structured parser representation; these original statements are preserved.

Who did the work

LLM only (people only prompted or submitted)

Models used

Anthropic Claude Opus 5.5

Thinking level / time

Max

Other LLMs

_No response_

Per-model versions, thinking and roles

Claude Opus 5.5 | version: claude-opus-5-5 via Claude Code CLI (exact build unknown) | thinking: max (Claude Code `/effort max`; all of the mathematical work ran at this setting, only the initial download of the prompt came before it) | time: about 30 minutes wall-clock in total (2026-10-19, roughly 12:19–12:50 CEST) | role: literature search, found the argument, wrote the proofs and exposition, self-audit

Model eligibility and verification evidence

Claude Opus 5.5 is one of the listed example models. No independent verification has been done: the proofs were checked only by the same model (self-review). No second model or person has checked them.

People and roles

@fersuarezz: prompted the model and submitted the issue; no mathematical contribution.

References

H. Bruhn, O. Schaudt, The journey of the union-closed sets conjecture, Graphs Combin. (2015), doi:10.1007/s00373-014-1515-0; arXiv:1309.3297. (Used: statement and separating reduction; Theorems 22–24; the discussion of improving the factor 2.)
V. Falgas-Ravry, Minimal weight in union-closed families, Electron. J. Combin. 18 (2011), #P95. (Used: Lemmas 1–2, the construction of the sets A_i.)
Y. Hu, On the union-closed sets conjecture, arXiv:1706.06167 (2017). (Used: Lemma 1 / Corollary 1, the domination lemma; Theorems 1–2 for comparison.)
J. Maßberg, The union-closed sets conjecture for small families, arXiv:1508.05718 (2015). (Used: Theorem 2.1 and inequality (13) for comparison; its summary of the known cases n ≤ 12 and m ≤ 50.)
R. Alweiss, B. Huang, M. Sellke, Improved lower bound for Frankl's union-closed sets conjecture, Electron. J. Combin. 31(3) (2024), P3.35; arXiv:2211.11731. (Used: Theorem 2 with Claim 4, the constant (3−√5)/2, which is input (E).)
Z. Chase, S. Lovett, Approximate union closed conjecture, arXiv:2211.11689 (2022). (Used: independent statement of the constant (3−√5)/2.)
W. Sawin, An improved lower bound for the union-closed set conjecture, arXiv:2211.11504. (Used: Theorem 1, an independent statement of the constant (3−√5)/2.)
J. Gilmer, A constant lower bound for the union-closed sets conjecture, arXiv:2211.09055 (2022). (Used: the entropy framework and the nondegeneracy convention.)
Y. Jiang, Entropy bounds and global couplings for union-closed families, arXiv:2609.08291 (2026). (Used: only the claimed constant 0.38288525 from the abstract; not verified.)
S. Cambie, Progress on the union-closed conjecture and offsprings in winter 2022–2023, arXiv:2306.12351. (Used: literature check for novelty only.)
we-solve contribution Math-415-A1 (TOP-55-A1, r1). (Read to avoid repeating it; not used in the proofs.)

Declarations

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