On set systems with strongly restricted intersections

Abstract

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 \(2t\le k\), then \(f_\alpha (n)\sim (t! n)^{1/t}\). More importantly, we show that for any constant \(c>0\), 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.