Heterochromatic Higher Order Transversals for Convex Sets
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In this short paper, we show that if {ℱ_n}_n ∈ℕ be a collection of families compact (r, R)-fat convex sets in ℝ^d and if every heterochromatic sequence with respect to {ℱ_n}_n ∈ℕ contains k+2 convex sets that can be pierced by a k-flat then there exists a family ℱ_m from the collection that can be pierced by finitely many k-flats. Additionally, we show that if {ℱ_n}_n ∈ℕ be a collection of families of compact convex sets in ℝ^d where each ℱ_n is a family of closed balls (axis parallel boxes) in ℝ^d and every heterochromatic sequence with respect to {ℱ_n}_n ∈ℕ contains 2 intersecting closed balls (boxes) then there exists a family ℱ_m from the collection that can be pierced by a finite number of points from ℝ^d. To complement the above results, we also establish some impossibility of proving similar results for other more general families of convex sets. Our results are a generalization of (ℵ_0,k+2)-Theorem for k-transversals of convex sets by Keller and Perles (Symposium on Computational Geometry 2022), and can also be seen as a colorful infinite variant of (p,q)-Theorems of Alon and Klietman (Advances in Mathematics 1992), and Alon and Kalai (Discrete Computational Geometry 1995).
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