Three-colorings of the plane with no monochromatic congruent triangle
Color every point of the plane with one of three colors. A congruent copy of a triangle $T$, its image under a rigid motion, is monochromatic if its three vertices receive the same color. The coloring is allowed to depend on $T$.
Graham posed the question in lectures at MSRI in August 2003, as part of his work on Euclidean Ramsey theory and as a much more restricted relative of the chromatic number of the plane, and conjectured that the answer is yes for every triangle.
Problem. Is it true that for every triangle $T$ in the plane there is a coloring of the plane with three colors in which no congruent copy of $T$ is monochromatic?
Source: Graham (MSRI lectures, August 2003); Problem 58 of The Open Problems Project (Demaine, Mitchell and O'Rourke).
Imported from topp.openproblem.net.
Classification
- Primary
- CO — Combinatorics
- Secondary
- MG
- Keywords
- Euclidean Ramsey theory, monochromatic triangle, plane coloring, The Open Problems Project
References
@incollection{graham2004handbook,
author = {Graham, R. L.},
title = {Euclidean {R}amsey theory},
booktitle = {Handbook of Discrete and Computational Geometry},
edition = {2nd},
editor = {Goodman, Jacob E. and O'Rourke, Joseph},
publisher = {CRC Press LLC},
address = {Boca Raton, FL},
year = {2004},
pages = {239--254},
}
@article{graham2004geocombinatorics,
author = {Graham, R. L.},
title = {Open problems in {E}uclidean {R}amsey theory},
journal = {Geocombinatorics},
volume = {13},
number = {4},
year = {2004},
pages = {165--177},
}
@article{orourke2004column46,
author = {O'Rourke, Joseph},
title = {Computational geometry column 46},
journal = {Internat. J. Comput. Geom. Appl.},
volume = {14},
number = {6},
year = {2004},
pages = {475--478},
note = {Also in SIGACT News 35(3):42--45 (2004)},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-01097, "Three-colorings of the plane with no monochromatic congruent triangle", clopen.solutions.