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We prove a tight colorful dimension-free Tverberg theorem asserting that for any pairwise disjoint k-point sets Q₁, , Qᵣ in the Euclidean space, there are a partition of the union Q₁ Qᵣ into r-point subsets P₁, , Pₖ and a ball of radius max₁ ₈ ₊₃₈₀₌ 㶁2r such that each Pᵢ shares with each Qⱼ exactly one point and the ball intersects the convex hulls of Pᵢ.
Barabanshchikova et al. (Sun,) studied this question.