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 $α=(a_1, \ldots, a_k)$, a collection $\mathcal F$ of subsets over an $n$ element set is an $α$-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 $f_α(n)$ 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 $a_t=1$ and $2t\leq k$, then $f_{α(n)} \sim {(t! n)}^{\frac 1 t}$. More importantly, we show that for any constant $c$, as the length $k$ goes larger, $f_α(n)$ is upper bounded by $O (n^c)$ for almost all $α$. Equivalently, no matter what $k$ is, there are only finitely many $α$ satisfying $f_α(n)=Ω(n^c)$. 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.