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.