Stabilities of the Kleitman diameter theorem

TL;DR

This paper advances the understanding of the Kleitman diameter theorem by characterizing extremal families and establishing a new stability result, thereby solving a recent open problem and enhancing the theorem's robustness.

Contribution

It provides a complete characterization of extremal families for Frankl's theorem and introduces a second stability result for Kleitman's diameter theorem.

Findings

Characterization of extremal families of Frankl's theorem

Establishment of a new stability result for Kleitman's theorem

Solves a recent open problem by Li and Wu

Abstract

Let $\mathcal{F}$ be a family of subsets of $[n]$. The diameter of $\mathcal{F}$ is the maximum size of symmetric differences among pairs of its members. Resolving a conjecture of Erd\H{o}s, Kleitman determined the maximum size of a family with fixed diameter, which states that a family with diameter $s$ has cardinality at most that of a Hamming ball of radius $s/2$. Specifically, if $\mathcal{F} \subseteq 2^{[n]}$ is a family with diameter $s$, then for $s=2d$, $|\mathcal{F}|\le \sum_{i=0}^d {n \choose i}$; for $s=2d+1$, $|\mathcal{F}|\le \sum_{i=0}^d {n \choose i} + {n-1 \choose d}$. This result is known as the Kleitman diameter theorem, which generalizes both the Katona union theorem and the Erd\H{o}s--Ko--Rado theorem. In 2017, Frankl provided a complete characterization of the extremal families of Kleitman's theorem and provided a stability result. In this paper, we determine the extremal families of Frankl's theorem and establish a further stability result of Kleitman's theorem. This solves a recent problem proposed by Li and Wu. Our findings constitute the second stability for the Kleitman diameter theorem.