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Eric Tressler
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One recent development is the discovery of "zebra-like" colorings that avoid an equilateral triangle; before, it was unknown whether the only 2-coloring to avoid an equilateral triangle is the obvious alternating strip coloring. The relevant paper is http://arxiv.org/abs/math.CO/0701940.

I have written on the subject. In http://www.math.ucsd.edu/~etressle/monotriangle.pdf I show a 3-coloring that avoids the 1,1,2 triangle. I will think about this for 2 colors. I believe it should be impossible, but I will update this later if I have more thoughts. It is easy to see (due to Soifer) that either a 1,1,1 or a 2,2,2 triangle imply that a 1,1,2 triangle exists (for 2 colors, again). It is also known that no two-coloring simultaneously avoids equilateral triangles of sides 1, 2, and 3.

The general conjecture is still wide open, as is the conjecture that 3 colors suffice to avoid any given triangle.

Eric Tressler
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