Polychromatic Coloring of Tuples in Hypergraphs

TL;DR

This work develops a systematic framework for polychromatic coloring of $t$-tuples in hypergraphs, focusing on geometric and bounded-VC-dimension settings. It establishes sharp bounds and tight relationships among $f_H(t,k)$, $f_H(1,x)$, and $\epsilon$-$t$-nets, including $f_{\mathcal{H}}(2,k)\le 3.7^{k}$ for planar disks and $f_{\mathcal{H}}(t,k)\le (\frac{4}{5ed^{3}})^{k}$ in higher dimensions with $t=\lfloor\frac{d+3}{2}\rfloor$, as well as $f_H(d{+}1,k)\le c^{k-1}$ for shrinkable hypergraphs of VC-dimension at most $d$. The paper also proves depth-based results ensuring large $(d{+}1)$-tuples exist, and derives inequalities linking $t$-tuple polychromatic bounds to vertex-coloring properties, plus specialized results for grids and pseudo-disks. Collectively, these results advance understanding of how to color higher-order tuples in geometric and abstract hypergraphs, with implications for cover-decomposability and epsilon-net decompositions in multiple dimensions.

Abstract

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.

Figures (2)

• Figure 1: (a) Shrinking $D$ to lose $q$. (b) Enlarging $D$ along the ray from $q$ to $c$.
• Figure 2: Illustration of the proof of Theorem \ref{['thm:disks-plane']}.

Theorems & Definitions (41)

• Definition 1: Shrinkability
• Definition 2: VC-dimension
• Definition 3: Depth
• Definition 4: $\epsilon$-$t$-Net
• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6