On set systems with strongly restricted intersections

Set systems with strongly restricted intersections, called \(\alpha \) $\alpha$ -intersecting families for a vector \(\alpha \) $\alpha$ , were introduced recently as a generalization of several well-studied intersecting families including the classical oddtown and eventown. Given a binary vector \(\alpha =(a_1, \ldots , a_k)\) $\alpha =(a_1,\dots ,a_k)$ , a collection \({\mathcal {F}}\) $\mathcal{F}$ of subsets over an n element set is an \(\alpha \) $\alpha$ -intersecting family modulo 2 if for each \(i=1,2,\ldots ,k\) $i=1,2,\dots ,k$ , all i-wise intersections of distinct members in \({\mathcal {F}}\) $\mathcal{F}$ have sizes with the same parity as \(a_i\) $a_i$ . Let \(f_\alpha (n)\) $f_{\alpha }(n)$ denote the maximum size of such a family. In this paper, we study the asymptotic behavior of \(f_\alpha (n)\) $f_{\alpha }(n)$ when n goes to infinity. We show that if t is the maximum integer such that \(a_t=1\) $a_t=1$ and \(2t\le k\) $2t\le k$ , then \(f_\alpha (n)\sim (t! n)^{1/t}\) $f_{\alpha }(n)\sim {(t!n)}^{1/t}$ . More importantly, we show that for any constant \(c>0\) $c>0$ , as the length k goes larger, \(f_\alpha (n)\) $f_{\alpha }(n)$ is upper bounded by \(O (n^c)\) $O(n^c)$ for almost all \(\alpha \) $\alpha$ . Equivalently, no matter what k is, there are only finitely many \(\alpha \) $\alpha$ satisfying \(f_\alpha (n)=\Omega (n^c)\) $f_{\alpha }(n)=\mathrm{\Omega }(n^c)$ . 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.