Bounded Fractional Intersecting Families are Linear in Size - Balachandran, Niranjan; Das, Shagnik; Sankarnarayanan, Brahadeesh (The Electronic journal of combinatorics, 2025)
Using the sunflower method, we show that if $θ \in (0,1) \cap \mathbb{Q}$ and $\mathcal{F}$ is a $O(n^{1/3})$-bounded $θ$-intersecting family over $[n]$, then $\lvert \mathcal{F} \rvert = O(n)$, and that if $\mathcal{F}$ is $o(n^{1/3})$-bounded, then $\lvert \mathcal{F} \rvert ≤ (\frac{3}{2} + o(1))n$. This partially solves a conjecture of Balachandran, Mathew and Mishra that any $θ$-intersecting family over $[n]$ has size at most linear in $n$, in the regime where we have no very large sets.