The Oddtown problem modulo a composite number

Abstract

A family of subsets $\mathcal{A}$ of an $n$-element set is called an $\ell$-Oddtown if the sizes of all sets are not divisible by $\ell$, but the sizes of pairwise intersections are divisible by $\ell$. Berlekamp and Graver showed that when $\ell$ is a prime, the maximum size of an $\ell$-Oddtown is $n$. Babai and Frankl extended this to prime powers, and asked whether the maximum size is still $n$ when $\ell$ is not a prime power, a question that was open even for $\ell=6$. For square-free composite moduli with $\omega$ distinct prime factors, the argument of Szegedy gives an upper bound of $\omega n-\omega\log_2 n$ on the size of an $\ell$-Oddtown. We answer the question of Babai and Frankl in the negative by constructing $\ell$-Oddtowns of size $\omega n-o(n)$, which shows that the leading term $\omega n$ cannot be improved. We also improve Szegedy's upper bound to $\omega n-(2\omega +\varepsilon)\log_2 n$ for most $\ell$ and $n$ using a combination of linear algebraic and Fourier-analytic arguments.