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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| Published in | The Electronic journal of combinatorics Vol. 32; no. 3 |
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| Main Authors | , , |
| Format | Journal Article |
| Language | English |
| Published |
22.08.2025
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| Online Access | Get full text |
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| Abstract | 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. |
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| AbstractList | 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. |
| Author | Balachandran, Niranjan Das, Shagnik Sankarnarayanan, Brahadeesh |
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| Snippet | 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... |
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| Title | Bounded Fractional Intersecting Families are Linear in Size |
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