# Characterizing 3-sets in Union-Closed Families

## TL;DR

This paper proves the 3-sets conjecture in union-closed families by classifying small families using polyhedral methods and integer programming, confirming the minimal number of 3-sets needed for Frankl's conjecture.

## Contribution

It provides a classification of 3-set families for small n and proves the 3-sets conjecture using polyhedral and integer programming techniques.

## Key findings

- Confirmed the 3-sets conjecture for all n ≥ 4.
- Classified families of 3-sets for n ≤ 9.
- Applied polyhedral interpretation of Poonen's Theorem.

## Abstract

A family of sets is union-closed (UC) if the union of any two sets in the family is also in the family. Frankl's UC sets conjecture states that for any nonempty UC family $\mathcal{F} \subseteq 2^{[n]}$ such that $\mathcal{F} \neq \left\{\emptyset\right\}$, there exists an element $i \in [n]$ that is contained in at least half the sets of $\mathcal{F}$. The 3-sets conjecture of Morris states that the smallest number of distinct 3-sets (whose union is an $n$-set) that ensure Frankl's conjecture is satisfied for any UC family that contains them is $ \lfloor{n/2\rfloor} + 1$ for all $n \geq 4$. For an UC family $\mathcal{A} \subseteq 2^{[n]}$, Poonen's Theorem characterizes the existence of weights on $[n]$ which ensure all UC families that contain $\mathcal{A}$ satisfy Frankl's conjecture, however the determination of such weights for specific $\mathcal{A}$ is nontrivial even for small $n$. We classify families of 3-sets on $n \leq 9$ using a polyhedral interpretation of Poonen's Theorem and exact rational integer programming. This yields a proof of the 3-sets conjecture.