On set systems with strongly restricted intersections

Set systems with strongly restricted intersections, called $\alpha$-intersecting families for a vector $\alpha$, were introduced recently as a generalization of several well-studied intersecting families including the classical oddtown and eventown. Given a binary vector $\alpha=(a_1, \ldots, a_k)$, a collection $\mathcal F$ of subsets over an $n$ element set is an $\alpha$-intersecting family modulo $2$ if for each $i=1,2,\ldots,k$, all $i$-wise intersections of distinct members in $\mathcal F$ have sizes with the same parity as $a_i$. Let $f_\alpha(n)$ denote the maximum size of such a family. In this paper, we study the asymptotic behavior of $f_\alpha(n)$ when $n$ goes to infinity. We show that if $t$ is the maximum integer such that $a_t=1$ and $2t\leq k$, then $f_{\alpha(n)} \sim {(t! n)}^{\frac 1 t}$. More importantly, we show that for any constant $c$, as the length $k$ goes larger, $f_\alpha(n)$ is upper bounded by $O (n^c)$ for almost all $\alpha$. Equivalently, no matter what $k$ is, there are only finitely many $\alpha$ satisfying $f_\alpha(n)=\Omega (n^c)$. 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.