# Coloring $d$ -Embeddable $k$ -Uniform Hypergraphs

Discrete & Computational Geometry, Oct 2014

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

This is a preview of a remote PDF: https://link.springer.com/content/pdf/10.1007%2Fs00454-014-9641-2.pdf

Carl Georg Heise, Konstantinos Panagiotou, Oleg Pikhurko, Anusch Taraz. Coloring $d$ -Embeddable $k$ -Uniform Hypergraphs, Discrete & Computational Geometry, 2014, 663-679, DOI: 10.1007/s00454-014-9641-2