Coloring Questions on Axis-Parallel Rectangles and Arithmetic Progressions

Abstract

We present an explicit family of hypergraphs with arbitrarily large uniformity and chromatic number that admit realizations in both geometric and number-theoretic settings. As an application, we give a new proof of a theorem of Chen, Pach, Szegedy, and Tardos. They showed that for any constants $c,k\ge1$, there exists a finite point set $P$ in the plane with the following property: for every coloring of $P$ with $c$ colors, there is an axis-parallel rectangle containing at least $k$ points, all of the same color. Their original proof is probabilistic; we present an explicit construction. Moreover, in the case $k=2$, we show that one can even realize a graph that has arbitrarily large girth and chromatic number simultaneously. We also answer a question of Pálvölgyi on coloring sets of integers with respect to certain finite arithmetic progressions. Finally, we give an application to coloring partially ordered sets.

Figures (5)

• Figure 1: The hypergraph $H^2_{2}$ and its realization by axis-parallel rectangles.
• Figure 2: Finding a monochromatic path edge in the $k=3$ case. Grey rectangles indicate a copy of $H_k^{c-1}$, red vertices indicate the elements in $B_j$ in each stage.
• Figure 3: The children of $v$ are placed below $v$ then shifted to the left to form a descending set.
• Figure 4: The two cases depending on how $v'$ and $w'$ were placed. The gray regions indicate the possible positions of $v'$ and $w'$ in the plane.
• Figure 5: The point set given by the van der Corput sequence for $V=\{1,3,7,8,10,15\}$. The grey rectangle captures $A\cap V$ for the arithmetic progression $A=3,5,7$.

Theorems & Definitions (21)

• Theorem 1.1: Chen, Pach, Szegedy, and Tardos chen2009delaunay
• Theorem 1.2
• Theorem 1.3
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Corollary 1.8
• Lemma 2.1
• proof
• Lemma 3.1