Coloring d-Embeddable k-Uniform Hypergraphs
Carl Georg Heise, Konstantinos Panagiotou, Oleg Pikhurko, Anusch Taraz

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.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
