Improved bound on symmetric differences of intersecting families

Abstract

For a family $\mathcal{F}$, it is called intersecting if $F\cap F^{\prime }\ne \varnothing$ for all $F,F^{\prime }\in \mathcal{F}$. We use $\mathcal{S}\mathcal{D}(\mathcal{F})=\{F△G:F,G\in \mathcal{F}\}$ to denote the family of symmetric differences of $\mathcal{F}$. In 2023, Frankl, Kiselev and Kupavskii conjectured that for any intersecting family $\mathcal{F}\subseteq \left(\genfrac{}{}{0ex}{}{[n]}{k}\right)$ with $n>10k$, the inequality $|\mathcal{S}\mathcal{D}(\mathcal{F})|\le {\sum }_{\ell =0}^{k-1}\left(\genfrac{}{}{0ex}{}{n-1}{2\ell }\right)$ holds. They further observed that a proof for the range $n>3k^2$ could likely be obtained via arguments similar to those in their earlier work, though no detailed derivation was given. In this paper, we establish the conjecture under the conditions $n\ge 100k\ln k$ and $k\ge 50$. We also determine the extremal families, which are precisely a certain class of stars. A concentration inequality plays a central role in the proof.