Balanced Sperner families via the topological Tverberg theorem

Chong Shangguan, Zixiang Xu, Yulin Yang

For every prime power $r\ge 2$, we show that any Sperner family $\mathcal{F}\subseteq 2^{[n]}$ with $|\mathcal{F}|\ge (r-1)n+1$ contains $r$ pairwise disjoint nonempty subfamilies wh…