The Open Problems Project

Next: Problem 59: Most Circular Partition of a Square

Previous: Problem 57: Chromatic Number of the Plane

Problem 58: Monochromatic Triangles


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.


Ron Graham, MSRI, August 2003.


Open. Ron Graham conjectures that the answer is yes for all triangles T.


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 in August 2003.

Partial and Related Results

See [O'R04] for further explanation.

Related Open Problems

Problem 57.


Ron Graham offers $50 for a solution.




combinatorial geometry

Entry Revision History

J. O’Rourke, 15 Aug. 2004.



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


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.


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.