A few new oddtown and eventown problems

Abstract

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

revisions from referee report