Bounded fractional intersecting families are linear in size

Using the sunflower method, we show that if θ ∈ (0,1) ∩ Q and F is a O(n1/3)-bounded θ-intersecting family over $[n]$, then F = O(n), and that if F is o(n1/3)-bounded, then F ≤ (3/2 + o(1))n. This partially solves a conjecture raised in (Balachandran et al., Electron J. Combin. 26 (2019), #P2.40) that any θ-intersecting family over $[n]$ has size at most linear in n, in the regime where we have no very large sets.