Using the sunflower method, we show linear size constraints for bounded theta-intersecting families over [n].
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 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.
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Balachandran et al. (2025) studied this question.