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Problem 58: Monochromatic Triangles
 Statement
 For any (planar) triangle T,
is there is a 3coloring 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
^{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.
 Gra04b

R. L. Graham.
Open problems in Euclidean Ramsey theory.
Geocombinatorics, XIII(4):165177, 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 239254. 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):475478, 2004.
Also in SIGACT News, 35(3):4245 (2004), Issue 132.
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The Open Problems Project  September 19, 2017