On the maximum size of $(a,b)$-town (mod $k$) families

Abstract

For integers $n \geq k \geq 2$ and $0 \leq a,b \leq k-1$, let $m_{k,n}(a,b)$ denote the maximum size of an $(a,b)$-town (mod $k$) family of an $n$-element set, a collection of subsets of whose cardinalities are congruent to $a$ modulo $k$ and whose pairwise intersections are congruent to $b$ modulo $k$. This notion generalizes the classical Oddtown and Eventown problems. We prove that $m_{k,n}(a,b)\leq n$ whenever $a\not\equiv b\pmod{k}$, thereby resolving a conjecture of Veselinov and Marinov. We also disprove another conjecture of theirs by showing that $m_{3,11}(2,2)>m_{3,11}(1,1)$. For the diagonal case $a\equiv b\pmod{k}$, we establish the general bound $m_{k,n}(a,a)\leq 2^{\lfloor n/2\rfloor}$ and completely determine when equality holds. We further obtain improved bounds and exact values in several special cases. The proofs combine characteristic-zero linear algebra with methods from coding theory and finite geometry.