Polychromatic Colorings of Geometric Hypergraphs via Shallow Hitting Sets

A range family ℛ is a family of subsets of ℝ^d, like all halfplanes, or all unit disks. Given a range family ℛ, we consider the m-uniform range capturing hypergraphs ℋ(V,ℛ,m) whose vertex-sets V are finite sets of points in ℝ^d with any m vertices forming a hyperedge e whenever e = V ∩ R for some R ∈ ℛ. Given additionally an integer k ≥ 2, we seek to find the minimum m = m_ℛ(k) such that every ℋ(V,ℛ,m) admits a polychromatic k-coloring of its vertices, that is, where every hyperedge contains at least one point of each color. Clearly, m_ℛ(k) ≥ k and the gold standard is an upper bound m_ℛ(k) = O(k) that is linear in k.
A t-shallow hitting set in ℋ(V,ℛ,m) is a subset S ⊆ V such that 1 ≤ |e ∩ S| ≤ t for each hyperedge e; i.e., every hyperedge is hit at least once but at most t times by S. We show for several range families ℛ the existence of t-shallow hitting sets in every ℋ(V,ℛ,m) with t being a constant only depending on ℛ. This in particular proves that m_ℛ(k) ≤ tk = O(k) in such cases, improving previous polynomial bounds in k. Particularly, we prove this for the range families of all axis-aligned strips in ℝ^d, all bottomless and topless rectangles in ℝ², and for all unit-height axis-aligned rectangles in ℝ².