Frankl’s Union-Closed Sets Conjecture

Abstract

This study explores structural and pattern-based factors influencing the validity of Frankl’s (Union-Closed Sets) Conjecture. By analyzing categorized families based on size, in- ternal structure, and symmetry, we assess how these characteristics impact the conjecture’s likelihood of holding. Particular focus is placed on families with high symmetry, dominant elements, or inherent union-closure. The results reveal consistent trends where certain configurations satisfy the conjecture trivially. These insights help pinpoint structural conditions that simplify the conjecture’s verification and highlight families that may present greater complexity.