Cutting Planes for Families Implying Frankl's Conjecture

TL;DR

This paper introduces a cutting-plane algorithm to identify families of sets that guarantee Frankl's conjecture, solving open questions and constructing counterexamples in the theory of union-closed families.

Contribution

It develops a novel algorithmic framework using cutting planes to find families implying Frankl's conjecture and addresses open structural questions in union-closed families.

Findings

Identified new families ensuring Frankl's conjecture holds.

Constructed a counterexample to Morris's 2006 conjecture.

Provided explicit weights for Poonen's Theorem conditions.

Abstract

We find previously unknown families of sets which ensure Frankl's conjecture holds for all families that contain them using an algorithmic framework. The conjecture states that for any nonempty union-closed (UC) family there exists an element of the ground set in at least half the sets of the considered UC family. Poonen's Theorem characterizes the existence of weights which determine whether a given UC family implies the conjecture for all UC families which contain it. We design a cutting-plane method that computes the explicit weights which imply the existence conditions of Poonen's Theorem. This method enables us to answer several open questions regarding structural properties of UC families, including the construction of a counterexample to a conjecture of Morris from 2006.