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.

In http://nucularpower.com/papers/monotriangle.pdf I show a 3-coloring that avoids the 1,1,2 triangle; I have been unable to find a 2-coloring that does so. 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.

**Edit, December 2016:** I have updated the link to my note. In case the link becomes outdated again, the 3-coloring that avoids a degenerate triangle is just the coloring shown below, where the upper hexagon illustrates how boundaries are colored.

[![Hexagonal coloring that avoids a monochromatic degenerate triangle][1]][1]


  [1]: https://i.sstatic.net/VEvt6.png