On the current state of the Union-closed sets conjecture

Introduction

The union-closed sets conjecture is a famous open problem in set theory.

A family of sets is said to be union-closed if the union of any two sets from the family also belongs to it.

Conjecture (Frankl, 1979). For every union-closed family containing a non-empty set there exists an element that belongs to at least half of the sets in the family.

Let’s switch to formal notation. Let $\mathcal{F}\subseteq\mathscr{P}(n)$ be a family of subsets of $[n]=\{1,2,\ldots,n\}$ and denote $m=|\mathcal{F}|$. For each $x\in[n]$ define the degree of $x$ in $\mathcal{F}$ as the number of sets containing $x$:

$$ d_x:=|\{i\in[m]:x\in F_i\}| $$

and the normalized degree (i.e., the fraction of sets the element belongs to) as $\tilde d_x=\frac{d_x}{m}$.

We can now reformulate the union-closed sets conjecture and state it using the introduced terminology.

Conjecture. Let $\mathcal{F}\subseteq\mathscr{P}(n)$ be a family of sets such that for any $A,B\in\mathcal{F}$ we have $A\cup B\in\mathcal{F}$. Then $$ \max_{i\in[n]}\tilde d_i\geq\frac{1}{2}\text{.} $$

Results so far

A breakthrough has been the proof of $\max_{i\in[n]}\tilde d_i\geq c$ for the constant $c=0.01$ by Justin Gilmer in 2022 arXiv:2211.09055v2 through an entropy-based approach. See the video below for a lecture by him outlining the discovery.

Since he uses entropy, he formulates the conjecture probabilistically:

Through subsequent papers, the constant has been improved to $c\approx0.3823455$ (arXiv:2212.12500v2). Evidence suggests it could be improved further, see the work of Liu from 2023.

Conclusion

I feel confident that progress will be made in the years to follow. It would be of great interest to see if a constant (or, in the best case, the complete) lower bound can be derived using some other approach, maybe even a purely combinatorial one.