The following problem related to the Hadwiger–Nelson problem was proposed by Ron Graham in R. L. Graham. Open problems in Euclidean Ramsey theory. Geocombinatorics, XIII(4):165–177, April 2004.
Problem. 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 $T$.
The most recent reference on the problem I have found is this: http://www.ist.tugraz.at/files/publications/geometry/ap-tcep-19.pdf
I am looking for other references related to this problem. I am mostly interested if it is still an open question whether 6 colors are enough for every triangle.