Sharp bounds on $k$-wise generalizations of oddtowns and eventowns

Abstract

For $\mathbf{\alpha }=({\alpha }_1,\dots ,{\alpha }_k)\in {\mathbb{F}}_2^k$, an $\mathbf{\alpha }$-town is a set family in which every $i$-wise intersection has parity ${\alpha }_i$. Denote by $f_{\mathbf{\alpha }}(n)$ the maximum size of an $\mathbf{\alpha }$-town on $[n]$. The classical oddtown and eventown problems study the cases $\mathbf{\alpha }=(1,0)$ and $(0,0)$, respectively. We determine the sharp asymptotics of $f_{\mathbf{\alpha }}(n)$ for all $\mathbf{\alpha }$, answering questions of Johnston--O'Neill and Wei--Zhang--Ge. We also study a symmetric variant $g_{\mathbf{\alpha }}(n)$, in which $i$-wise intersection sizes $|F_1\cap \cdots \cap F_i|$ are replaced by $i$-wise intersection-union sizes $|F_1\cap \cdots \cap F_i|+|F_1\cup \cdots \cup F_i|$.

17 pages