For any (planar) triangle \(T\), is there is a \(3\)-coloring of the (infinite) plane with no monochromatic copy of \(T\)?
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 \(T\).
| Author | Date Published | Prize | Paid by |
| Ron Graham | August 2003 | $50 | Ron Graham |
Ron Graham [1] [2] conjectures that the answer is yes for all triangles \(T\).
[1] R. L. Graham. Open problems in Euclidean Ramsey theory. Geocombinatorics, XIII(4):165–177, April 2004.
[2] 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.
[3] 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.
[4] The Open Problems Project, Erik D. Demaine, Joseph S. B. Mitchell, Joseph O’Rourke https://topp.openproblem.net/
3-Coloring; Monochromatic Triangles; Money; Combinatorial Geometry