---
tag: 4628Z96WVZ
kind: conjecture
name: Union-closed sets conjecture
msc: [05D05]
uses: [0C21Q694AG, 0X1GSWNBN1]
about: [0C21Q694AG]
sources:
  - url: https://en.wikipedia.org/wiki/Union-closed_sets_conjecture
same_as:
  wikidata: Q2993338
  wikipedia: Union-closed sets conjecture
history:
  posed_by: "Péter Frankl"
  first_stated: "1979"
---

## Statement

Let $\mathcal{F}$ be a finite [[family|0C21Q694AG]] of finite sets that contains at least one nonempty set.
Suppose that $\mathcal{F}$ is union-closed: $A \cup B \in \mathcal{F}$ for all $A, B \in \mathcal{F}$, where $\cup$ is the [[union|0X1GSWNBN1]].
Then there is an element $x$ that belongs to at least $|\mathcal{F}| / 2$ of the sets in $\mathcal{F}$.

## Remarks

The conjecture is attributed to Péter Frankl and is also called Frankl's conjecture.
It is open.
In 2022 Justin Gilmer proved that some element lies in at least $0.01 \, |\mathcal{F}|$ of the sets, the first bound by a positive constant fraction.
Several authors then improved the constant to about $0.38$.
The conjecture is known when $\mathcal{F}$ contains a set with one or two elements: such an element, or one of the two elements, works.

Example: $\mathcal{F} = \set{\emptyset, \set{1}, \set{2}, \set{1, 2}}$ is union-closed, and the element $1$ lies in $2 = |\mathcal{F}|/2$ of its sets, so the bound $1/2$ cannot be improved.
The nonempty hypothesis is needed: $\mathcal{F} = \set{\emptyset}$ is union-closed, but no element lies in any of its sets.

## Explanations

### L1 human

Take a collection of sets in which the union of any two sets of the collection is again in the collection.
The conjecture says that some single element appears in at least half of the sets.
It is very easy to state, yet it resisted all attempts for decades, and only recently was a constant fraction proved.