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Problem 58: Monochromatic Triangles

Statement
For any (planar) triangle T, is there is a 3-coloring of the (infinite) plane with no monochromatic copy of T? We imagine congruent copies of T moved around the plane via rigid motions, and seek a spot where T is monochromatic. T is monochromatic if its three vertices are painted the same color, by virtue of lying on points of the plane painted that color. Note that the coloring in the question may depend on the given triangle T.
Origin
Ron Graham, MSRI, August 2003.
Status/Conjectures
Open. Ron Graham conjectures that the answer is YES for all triangles T.
Motivation
The question of the chromatic number of the Euclidean plane $ \mathbb {E}$2 has been unresolved for over fifty years (Problem 57). This problem is an interesting, much more restricted variant, posed by Ron Graham as part of his ``Geometric Ramsey Theory"'' investigation [Gra04a] [Gra04b] at his MSRI lectures in August 2003.
Partial and Related Results
See [O'R04] for further explanation.
Related Open Problems
Problem 57.
Reward
Ron Graham offers $50 for a solution.
Appearances
[O'R04]
Categories
combinatorial geometry
Entry Revision History
J. O'Rourke, 15 Aug. 2004.

Bibliography

Gra04b
R. L. Graham.
Open problems in Euclidean Ramsey theory.
Geocombinatorics, XIII(4):165-177, April 2004.

Gra04a
R. L. Graham.
Euclidean Ramsey theory.
In Jacob E. Goodman and Joseph O'Rourke, editors, Handbook of Discrete and Computational Geometry, chapter 11, pages 239-254. CRC Press LLC, Boca Raton, FL, 2nd edition, 2004.

O'R04
Joseph O'Rourke.
Computational geometry column 46.
Internat. J. Comput. Geom. Appl., 14(6):475-478, 2004.
Also in SIGACT News, 35(3):42-45 (2004), Issue 132.


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Next: Problem 59: Most Circular Up: The Open Problems Project Previous: Problem 57: Chromatic Number
The Open Problems Project - December 04, 2015