A class of inequalities for intersection-closed set systems

TL;DR

This paper proves a class of inequalities that support Frankl's conjecture, which states that in any intersection-closed set family, there exists an element in at most half the sets, unless the family is a singleton.

Contribution

The paper introduces and proves a new class of inequalities that imply Frankl's conjecture for intersection-closed set systems.

Findings

Validated a class of inequalities supporting Frankl's conjecture

Provided a mathematical framework for the conjecture's proof

Enhanced understanding of the structure of intersection-closed families

Abstract

Let $N$ be a finite set and $\mathcal{F}$, an intersection-closed family of subsets. Frankl conjectured that there always exists an element in $N$ which is contained in at most half the number of sets in $\mathcal{F}$ unless $\mathcal{F} =\{E\}$. We prove the validity of a class of inequalities which imply Frankl's conjecture.