Tverberg–Vrećica conjecture

Let $0 \le k < d$ be integers, and let $r_0, \dots, r_k$ be positive integers. Suppose that $P_0, \dots, P_k$ are sets of points in $\mathds{R}^d$ satisfying

$$|P_i|=(r_i-1)(d-k+1)+1$$

for each $i$. Tverberg–Vrećica conjecture. There exists a $k$-dimensional flat $V\subseteq\mathds{R}^d$ and, for each $i$, a partition of $P_i$ into $r_i$ parts such that the convex hull of every part of every partition intersects $V$. This conjecture generalizes the central transversal theorem in the same way that Tverberg's theorem generalizes Rado's centerpoint theorem. Its topological and colorful generalizations are known when all $r_i$ are powers of the same prime $p$ and $p(d-k)=p\cdot\operatorname{codim}_{\mathds{R}}V$ is even; it remains open otherwise.