Frankl’s union-closed sets conjecture

Prove Frankl’s union-closed sets conjecture asserting that every finite union-closed family of sets contains an element that belongs to at least half of the sets.

Background

The paper leverages entropy methods stemming from recent progress on the union-closed sets conjecture. Although constant-factor advances have been made (e.g., improving the fraction from 1/100 to approximately 0.382), the original conjecture remains unresolved.

This conjecture provides methodological inspiration for the paper’s sparsification techniques via entropy and VC-dimension arguments.