Discoveries / combinatorics

Babai and Frankl's Oddtown Question for Composite Moduli

An $\ell$-Oddtown is a family of subsets of an $n$-element set whose set sizes are not divisible by $\ell$ while all pairwise intersection sizes are. Berlekamp and Graver showed the maximum size is $n$ for prime $\ell$, Babai and Frankl extended this to prime powers and asked whether $n$ still holds for other moduli, a question open even for $\ell = 6$. Bukh, Chao and Zheng answer it negatively with an explicit superlinear construction, complemented by new upper bounds.