Bounded Fractional Intersecting Families are Linear in Size

Using the sunflower method, we show that if $\theta \in (0,1) \cap \mathbb{Q}$ and $\mathcal{F}$ is a $O(n^{1/3})$-bounded $\theta$-intersecting family over $[n]$, then $\lvert \mathcal{F} \rvert = O(n)$, and that if $\mathcal{F}$ is $o(n^{1/3})$-bounded, then $\lvert \mathcal{F} \rvert \leq (\frac{...

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Bibliographic Details
Published inThe Electronic journal of combinatorics Vol. 32; no. 3
Main Authors Balachandran, Niranjan, Das, Shagnik, Sankarnarayanan, Brahadeesh
Format Journal Article
LanguageEnglish
Published 22.08.2025
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Summary:Using the sunflower method, we show that if $\theta \in (0,1) \cap \mathbb{Q}$ and $\mathcal{F}$ is a $O(n^{1/3})$-bounded $\theta$-intersecting family over $[n]$, then $\lvert \mathcal{F} \rvert = O(n)$, and that if $\mathcal{F}$ is $o(n^{1/3})$-bounded, then $\lvert \mathcal{F} \rvert \leq (\frac{3}{2} + o(1))n$. This partially solves a conjecture of Balachandran, Mathew and Mishra that any $\theta$-intersecting family over $[n]$ has size at most linear in $n$, in the regime where we have no very large sets.
ISSN:1077-8926
1077-8926
DOI:10.37236/12900