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

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