On set systems with strongly restricted intersections

Set systems with strongly restricted intersections, called α-intersecting families for a vector α, were introduced recently as a generalization of several well-studied intersecting families including the classical oddtown and eventown. Given a binary vector α=(a₁, …, aₖ), a collection F of subsets over an n element set is an α-intersecting family modulo $2$ if for each i=1,2,…,k, all i-wise intersections of distinct members in F have sizes with the same parity as aᵢ. Let f_α(n) denote the maximum size of such a family. In this paper, we study the asymptotic behavior of f_α(n) when n goes to infinity. We show that if t is the maximum integer such that aₜ=1 and 2t≤ k, then fα(n) ~ (t! n)1 t. More importantly, we show that for any constant c, as the length k goes larger, f_α(n) is upper bounded by O (nᶜ) for almost all α. Equivalently, no matter what k is, there are only finitely many α satisfying f_α(n)=Ω (nᶜ). This answers an open problem raised by Johnston and O'Neill in 2023. All of our results can be generalized to modulo p setting for any prime p smoothly.