Colorful Tverberg conjecture (general case)

Establish the colorful Tverberg conjecture in full generality: For any pairwise disjoint k-point sets Q_1, ..., Q_{d+1} in the Euclidean space R^d, construct a partition of Q_1 ∪ ... ∪ Q_{d+1} into k subsets P_1, ..., P_k of size (d+1) such that each P_i contains exactly one point from each Q_j and the convex hulls of P_1, ..., P_k have a nonempty common intersection.

Background

Tverberg's theorem asserts that any (k−1)(d+1)+1 points in R^d can be partitioned into k parts whose convex hulls intersect, and the colorful variant strengthens this by requiring transversals across designated color classes. The colorful Tverberg conjecture posits that for d+1 color classes of k points each in R^d, there exists a partition into k colorful sets (each containing exactly one point from each color class) whose convex hulls intersect.

Despite extensive paper and progress on Tverberg-type results, the colorful Tverberg conjecture has resisted a complete solution and is recognized as a central unsolved problem in discrete geometry. The paper references its open status while focusing on dimension-free variations and algorithmic aspects.