A few new oddtown and eventown problems

Given a vector $\alpha =({\alpha }_1,\dots ,{\alpha }_k)\in {\mathbb{F}}_2^k$, we say a collection of subsets $\mathcal{F}$ satisfies $\alpha$-intersection pattern modulo $2$ if all $i$-wise intersections consisting of $i$ distinct sets from $\mathcal{F}$ have size ${\alpha }_i(\mathrm{m}\mathrm{o}\mathrm{d}2)$. In this language, the classical oddtown and eventown problems correspond to vectors $\alpha =(1,0)$ and $\alpha =(0,0)$ respectively. In this paper, we determine the largest such set families of subsets on a $n$-element set with $\alpha$-intersection pattern modulo $2$ for all $\alpha \in {\mathbb{F}}_2^3$ and all $\alpha \in {\mathbb{F}}_2^4$ asymptotically. Lastly, we consider the corresponding problem with restrictions modulo $3$.