On set systems with strongly restricted intersections

Set systems with strongly restricted intersections, called -intersecting families for a vector, were introduced recently as a generalization of several well-studied intersecting families including the classical oddtown and eventown. Given a binary vector = (a₁, , aₖ), a collection F of subsets over an n element set is an -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_ (n) denote the maximum size of such a family. In this paper, we study the asymptotic behavior of f_ (n) when n goes to infinity. We show that if t is the maximum integer such that aₜ=1 and 2t k, then f (₍) (t! n) ^ 1 t. More importantly, we show that for any constant c, as the length k goes larger, f_ (n) is upper bounded by O (nᶜ) for almost all. Equivalently, no matter what k is, there are only finitely many satisfying f_ (n) = (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.