A range family $\mathcal{R}$ is a family of subsets of $\mathbb{R}^d$, like all halfplanes, or all unit disks. Given a range family $\mathcal{R}$, we consider the $m$-uniform range capturing hypergraphs $\mathcal{H}(V,\mathcal{R},m)$ whose vertex-sets $V$ are finite sets of points in $\mathbb{R}^d$ with any $m$ vertices forming a hyperedge $e$ whenever $e = V \cap R$ for some $R \in \mathcal{R}$. Given additionally an integer $k \geq 2$, we seek to find the minimum $m = m_{\mathcal{R}}(k)$ such that every $\mathcal{H}(V,\mathcal{R},m)$ admits a polychromatic $k$-coloring of its vertices, that is, where every hyperedge contains at least one point of each color. Clearly, $m_{\mathcal{R}}(k) \geq k$ and the gold standard is an upper bound $m_{\mathcal{R}}(k) = O(k)$ that is linear in $k$. A $t$-shallow hitting set in $\mathcal{H}(V,\mathcal{R},m)$ is a subset $S \subseteq V$ such that $1 \leq |e \cap S| \leq t$ 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 $\mathcal{R}$ the existence of $t$-shallow hitting sets in every $\mathcal{H}(V,\mathcal{R},m)$ with $t$ being a constant only depending on $\mathcal{R}$. This in particular proves that $m_{\mathcal{R}}(k) \leq tk = O(k)$ in such cases, improving previous polynomial bounds in $k$. Particularly, we prove this for the range families of all axis-aligned strips in $\mathbb{R}^d$, all bottomless and topless rectangles in $\mathbb{R}^2$, and for all unit-height axis-aligned rectangles in $\mathbb{R}^2$.

翻译：范围族$\mathcal{R}$是$\mathbb{R}^d$中子集构成的族，例如所有半平面或所有单位圆盘。给定范围族$\mathcal{R}$，我们考虑$m$均匀范围捕获超图$\mathcal{H}(V,\mathcal{R},m)$，其顶点集$V$是$\mathbb{R}^d$中有限点集，当$e = V \cap R$（其中$R \in \mathcal{R}$）时，任意$m$个顶点构成超边$e$。给定整数$k \geq 2$，我们寻求最小$m = m_{\mathcal{R}}(k)$使得每个$\mathcal{H}(V,\mathcal{R},m)$的顶点存在多色$k$着色，即每个超边包含每种颜色的至少一个点。显然$m_{\mathcal{R}}(k) \geq k$，且黄金标准是线性于$k$的上界$m_{\mathcal{R}}(k) = O(k)$。$\mathcal{H}(V,\mathcal{R},m)$中的$t$浅层击中集是子集$S \subseteq V$，使得每个超边$e$满足$1 \leq |e \cap S| \leq t$；即$S$恰好击中每个超边至少一次但至多$t$次。我们证明对于若干范围族$\mathcal{R}$，每个$\mathcal{H}(V,\mathcal{R},m)$均存在$t$浅层击中集，其中$t$为仅依赖于$\mathcal{R}$的常数。这特别证明了在此类情形下$m_{\mathcal{R}}(k) \leq tk = O(k)$，改进了先前关于$k$的多项式界限。具体而言，我们为$\mathbb{R}^d$中所有轴向对齐带状、$\mathbb{R}^2$中所有无底无顶矩形以及$\mathbb{R}^2$中所有单位高轴向对齐矩形的范围族证明了该结论。