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arXiv · 2402.14981

Bounded fractional intersecting families are linear in size

Niranjan Balachandran·Shagnik Das·Brahadeesh Sankarnarayanan
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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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Niranjan Balachandran, Shagnik Das, Brahadeesh Sankarnarayanan. 2024-02-22. Bounded fractional intersecting families are linear in size. https://doi.org/10.37236/12900

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