Balanced Sperner families via the topological Tverberg theorem

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 whose unions are all equal and whose intersections are all equal. For $r=2$, this confirms a conjecture of Heged\"{u}s, with the sharp threshold $n+1$. In this purely combinatorial problem, our proof combines a multilinear polynomial method, a continuity argument, and the topological Tverberg theorem.