On the Chromatic Number of Subsets of the Euclidean Plane

@article{Axenovich2014OnTC,
  title={On the Chromatic Number of Subsets of the Euclidean Plane},
  author={M. Axenovich and JiHyeok Choi and Michelle A. Lastrina and T. McKay and J. Smith and B. Stanton},
  journal={Graphs and Combinatorics},
  year={2014},
  volume={30},
  pages={71-81}
}
The chromatic number of a subset of the real plane is the smallest number of colors assigned to the elements of that set such that no two points at distance 1 receive the same color. It is known that the chromatic number of the plane is at least 4 and at most 7. In this note, we determine the bounds on the chromatic number for several classes of subsets of the plane such as extensions of the rational plane, sets in convex position, infinite strips, and parallel lines. 
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References

SHOWING 1-10 OF 30 REFERENCES
The chromatic number of the plane: The bounded case
  • C. Kruskal
  • Computer Science, Mathematics
  • J. Comput. Syst. Sci.
  • 2008
TLDR
This paper gives tight bounds for two and three coloring regions bounded by circles, rectangles, and regular polygons, and shows how large a region can be and still be two or three colorable, and how complicated the colorings need to be. Expand
A lower bound on the number of unit distances between the vertices of a convex polygon
TLDR
It is proved that for every n ⩾ 4 there is a convex n-gon such that the vertices of 2n − 7 vertex pairs are one unit of distance apart. Expand
Distances Realized by Sets Covering the Plane*
A proof is given of the (known) result that, if real n-dimensional Euclidean space R” is covered by any n + 1 sets, then at least one of these sets is such that each distance d (0 < d < a)) isExpand
Multiplicities of Interpoint Distances in Finite Planar Sets
TLDR
This work reviews known partial results for this and other open questions on multiple occurrences of the same interpoint distance in finite planar subsets and proves some new results for small n. Expand
Distances Realized by Sets Covering the Plane
  • D. R. Woodall
  • Computer Science, Mathematics
  • J. Comb. Theory, Ser. A
  • 1973
Abstract A proof is given of the (known) result that, if real n-dimensional Euclidean space Rn is covered by any n + 1 sets, then at least one of these sets is such that each distance d(0
The Mathematical Coloring Book: Mathematics of Coloring and the Colorful Life of its Creators
Merry-Go-Round.- A Story of Colored Polygons and Arithmetic Progressions.- Colored Plane.- Chromatic Number of the Plane: The Problem.- Chromatic Number of the Plane: An Historical Essay.-Expand
All Unit-Distance Graphs of Order 6197 Are 6-Colorable
  • D. Pritikin
  • Computer Science, Mathematics
  • J. Comb. Theory, Ser. B
  • 1998
TLDR
It is shown that the vertices of the resulting graph can be properly 6-colored and created by considering any 6197 or fewer points in the plane. Expand
Additive K-colorable extensions of the rational plane
TLDR
It is shown for example, that if F2 is 2-colorable and if √2∉F, then F2 contains no regular polygon except for the square, which generalizes the classical result known for the rational plane. Expand
The Maximum Number of Times the Same Distance Can Occur among the Vertices of a Convex n-gon Is O(n log n)
TLDR
A short proof of Z. Furedi's theorem (1990, J. Combin. Theory Ser. A55, 316?320) stated in the title is presented. Expand
A colour problem for infinite graphs and a problem in the theory of relations
Our original proof was simplified by SZEKERES. Later, a simple proof, based on TychonofI’s theorem that the Cartesian product of a family of compact sets is compact, was indicated by RABSOX and A.Expand
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