Coloring d-Embeddable k-Uniform Hypergraphs

TL;DR

This paper investigates the chromatic numbers of hypergraphs embeddable in Euclidean space, providing bounds based on the dimension and uniformity, extending concepts related to the Four Color Theorem.

Contribution

It establishes new bounds on the chromatic numbers of hypergraphs embeddable in R^d, generalizing coloring problems in higher-dimensional spaces.

Findings

Existence of hypergraphs with logarithmic chromatic number in certain embeddings

Upper bounds on chromatic number for hypergraphs in H(d,d)

Extension of coloring bounds from graphs to hypergraphs in Euclidean spaces

Abstract

This paper extends the scenario of the Four Color Theorem in the following way. Let H(d,k) be the set of all k-uniform hypergraphs that can be (linearly) embedded into R^d. We investigate lower and upper bounds on the maximum (weak and strong) chromatic number of hypergraphs in H(d,k). For example, we can prove that for d>2 there are hypergraphs in H(2d-3,d) on n vertices whose weak chromatic number is Omega(log n/log log n), whereas the weak chromatic number for n-vertex hypergraphs in H(d,d) is bounded by O(n^((d-2)/(d-1))) for d>2.