A hypergraph $H$ consists of a set $V$ of vertices and a set $E$ of hyperedges that are subsets of $V$. A $t$-tuple of $H$ is a subset of $t$ vertices of $V$. A $t$-tuple $k$-coloring of $H$ is a mapping of its $t$-tuples into $k$ colors. A coloring is called $(t,k,f)$-polychromatic if each hyperedge of $E$ that has at least $f$ vertices contains tuples of all the $k$ colors. Let $f_H(t,k)$ be the minimum $f$ such that $H$ has a $(t,k,f)$-polychromatic coloring. For a family of hypergraphs $\cal{H}$ let $f_{\cal{H}}(t,k)$ be the maximum $f_H(t,k)$ over all hypergraphs $H$ in $\cal{H}$. We present several bounds on $f_{\cal{H}}(t,k)$ for $t\ge 2$. - Let $\cal{H}$ be the family of hypergraphs $H$ that is obtained by taking any set $P$ of points in $\Re^2$, setting $V:=P$ and $E:=\{d\cap P\colon d\text{ is a disk in }\Re^2\}$. We prove that $f_\cal{H}(2,k)\le 3.7^k$, that is, the pairs of points (2-tuples) can be $k$-colored such that any disk containing at least $3.7^k$ points has pairs of all colors. - For the family $\mathcal{H}$ of shrinkable hypergraphs of VC-dimension at most $d$ we prove that $ f_\cal{H}(d{+}1,k) \leq c^k$ for some constant $c=c(d)$. We also prove that every hypergraph with $n$ vertices and with VC-dimension at most $d$ has a $(d{+}1)$-tuple $T$ of depth at least $\frac{n}{c}$, i.e., any hyperedge that contains $T$ also contains $\frac{n}{c}$ other vertices. - For the relationship between $t$-tuple coloring and vertex coloring in any hypergraph $H$ we establish the inequality $\frac{1}{e}\cdot tk^{\frac{1}{t}}\le f_H(t,k)\le f_H(1,tk^{\frac{1}{t}})$. For the special case of $k=2$, we prove that $t+1\le f_H(t,2)\le\max\{f_H(1,2), t+1\}$; this improves upon the previous best known upper bound. - We generalize some of our results to higher dimensions, other shapes, pseudo-disks, and also study the relationship between tuple coloring and epsilon nets.

翻译：超图$H$由顶点集$V$和超边集$E$构成，其中超边是$V$的子集。$H$的一个$t$-元组是$V$中$t$个顶点的子集。$H$的$t$-元组$k$-着色是指将其$t$-元组映射到$k$种颜色。若$E$中每个至少包含$f$个顶点的超边都包含所有$k$种颜色的元组，则称该着色为$(t,k,f)$-多色着色。令$f_H(t,k)$为使得$H$存在$(t,k,f)$-多色着色的最小$f$值。对于超图族$\cal{H}$，令$f_{\cal{H}}(t,k)$为所有$\cal{H}$中超图$H$对应的$f_H(t,k)$的最大值。本文针对$t\ge 2$给出了$f_{\cal{H}}(t,k)$的若干界。- 设$\cal{H}$为通过以下方式构造的超图族：取$\Re^2$中任意点集$P$，令$V:=P$，$E:=\{d\cap P\colon d\text{是}\Re^2\text{中的圆盘}\}$。我们证明$f_\cal{H}(2,k)\le 3.7^k$，即点对（2-元组）可进行$k$-着色，使得任何包含至少$3.7^k$个点的圆盘都包含所有颜色的点对。- 对于VC维至多为$d$的可收缩超图族$\mathcal{H}$，我们证明存在常数$c=c(d)$使得$f_\cal{H}(d{+}1,k) \leq c^k$。同时证明：任意具有$n$个顶点且VC维至多为$d$的超图都存在深度至少为$\frac{n}{c}$的$(d{+}1)$-元组$T$，即任何包含$T$的超边还包含$\frac{n}{c}$个其他顶点。- 对于任意超图$H$中$t$-元组着色与顶点着色的关系，我们建立了不等式$\frac{1}{e}\cdot tk^{\frac{1}{t}}\le f_H(t,k)\le f_H(1,tk^{\frac{1}{t}})$。针对$k=2$的特殊情况，我们证明$t+1\le f_H(t,2)\le\max\{f_H(1,2), t+1\}$，该结果改进了先前已知的最佳上界。- 我们将部分结果推广到更高维度、其他几何形状、伪圆盘情形，并研究了元组着色与epsilon网之间的关系。