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 $O(n^{1\mathrm{/}3})$-bounded $\theta$-intersecting family over $[n]$, then $\mid \mathcal{F}\mid =O(n)$, and that if $\mathcal{F}$ is $o(n^{1\mathrm{/}3})$-bounded, then $\mid \mathcal{F}\mid \le (\frac{3}{2}+o(1))n$. 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.