A class of inequalities for intersection-closed set systems

TL;DR

The paper addresses the Frankl conjecture for intersection-closed set systems by developing a counting-inequality framework built on discarding sets $D_i$ and excluded supersets $H_i^A$, guided by the Helly property to avoid double counting. A central main theorem establishes a nondecreasing bound sequence $t^i$, starting from $t^0=2^{n-1}$, that leads to the global bound $|F|\ge |F_{n-1}|+|F_n|$, thereby proving Frankl's conjecture and characterizing equality cases. A stronger result shows that the rare element occurs in exactly half the sets only when the family $F$ is a Boolean algebra, providing a complete structural characterization. The work offers a constructive combinatorial approach with potential applicability to related intersection-closed or lattice-type set-system questions and long-standing conjectures.

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.

Theorems & Definitions (14)

• Lemma 4.1
• proof
• Lemma 4.2
• proof
• Lemma 4.3
• proof
• Theorem 5.1
• proof
• Corollary 6.1: Frankl's conjecture
• proof