Sharp bounds on k-wise generalizations of oddtowns and eventowns

For $\boldsymbol{\alpha} = (\alpha_1, \dots, \alpha_k) \in \mathbb{F}_2^k$, an $\boldsymbol{\alpha}$-town is a set family in which every $i$-wise intersection has parity $\alpha_i$. Denote by $f_{\boldsymbol{\alpha}}(n)$ the maximum size of an $\boldsymbol{\alpha}$-town on $[n]$. The classical oddtown and eventown problems study the cases $\boldsymbol{\alpha} = (1, 0)$ and $(0, 0)$, respectively. We determine the sharp asymptotics of $f_{\boldsymbol{\alpha}}(n)$ for all $\boldsymbol{\alpha}$, answering questions of Johnston--O'Neill and Wei--Zhang--Ge. We also study a symmetric variant $g_{\boldsymbol{\alpha}}(n)$, in which $i$-wise intersection sizes $|F_1 \cap \dots \cap F_i|$ are replaced by $i$-wise intersection-union sizes $|F_1 \cap \dots \cap F_i| + |F_1 \cup \dots \cup F_i|$.