Open problem, Combinatorics and discrete mathematics, posed 1979

Union-closed sets conjecture (Frankl)

Open

Let $\mathcal{F}$ be a finite family of finite sets, not consisting solely of the empty set $\mathcal{F} \neq \{\emptyset\}$ , that is union-closed (for all $A, B \in \mathcal{F}$ , their union $A \cup B \in \mathcal{F}$ ). Then there exists an element $x$ in the ground set $\bigcup_{A \in \mathcal{F}} A$ that belongs to at least half of the sets in $\mathcal{F}$ , i.e., $\#\{A \in \mathcal{F} : x \in A\} \ge \frac{1}{2} |\mathcal{F}|$ .

Research frontier as of 2026

As of 2026, Frankl's $\frac{1}{2}$ conjecture remains open, with the best general constant lower bound standing just above $\frac{3 - \sqrt{5}}{2} \approx 0.38197$ (at $\approx 0.38234$ , Yu, 2023). Crucially, Sawin and Alweiss–Huang–Sellke showed that $\frac{3 - \sqrt{5}}{2}$ is the exact barrier for Gilmer's single-distribution i.i.d. entropy method because approximate union-closed distributions on products of two-point spaces achieve equality there. Reaching $\frac{1}{2}$ requires exploiting the exact combinatorial integrality of $\mathcal{F}$ beyond first-order entropy comparisons.

Best known results

Every finite union-closed family $\mathcal{F} \neq \{\emptyset\}$ has an element of frequency at least $\frac{3 - \sqrt{5}}{2} \approx 0.38197$ (Alweiss–Huang–Sellke, Chase–Lovett, Pebody, and Sawin, 2022, following Gilmer), refined to $\approx 0.38234$ (2023).
The full $\frac{1}{2}$ conjecture holds for all union-closed families on ground sets of size $n \le 12$ and families with $|\mathcal{F}| \le 50$ (Bošnjak–Marković, 2008; Roberts–Simpson, 2010).

Tools and where they stop

Tool	Achieved	Where it stops
Shannon entropy of independent random subsets (Gilmer's method)	Samples $A, B \in \mathcal{F}$ independently so $A \cup B \in \mathcal{F}$ and uses submodularity of conditional entropy to show $H(A \cup B) > H(A)$ whenever all marginal probabilities are below $\frac{3 - \sqrt{5}}{2}$ , forcing a contradiction.	For two-state product distributions where each coordinate is $1$ with probability $p > \frac{3 - \sqrt{5}}{2}$ , the binary entropy satisfies $h(2p - p^2) \le h(p)$ , blocking single-distribution entropy bounds at $\frac{3 - \sqrt{5}}{2}$ .
Weight-averaging and local configuration reductions	Proves the $\frac{1}{2}$ bound whenever $\mathcal{F}$ contains small sets of size $1$ or $2$ , or when $\|\mathcal{F}\|$ is large relative to the ground set size $n$ .	Fails when the minimum non-empty set size in $\mathcal{F}$ grows with $n$ and $\|\mathcal{F}\|$ is polynomial or sub-exponential in $n$ .

Open questions

Does every finite union-closed family $\mathcal{F} \neq \{\emptyset\}$ contain an element belonging to at least $\frac{1}{2} |\mathcal{F}|$ sets?
Can multi-sample or higher-order information-theoretic inequalities break the $\frac{3 - \sqrt{5}}{2}$ barrier by a large margin toward $\frac{1}{2}$ ?

References

Justin Gilmer (2022). A constant lower bound for the union-closed sets conjecture · arXiv:2211.09055 [preprint, not peer-reviewed]
Henning Bruhn, Oliver Schaudt (2015). The journey of the union-closed sets conjecture · DOI:10.1007/s00373-014-1515-0