An old problem (possibly due to Erdős and Graham?): given a triangle $T$ and a two-coloring of the plane, does there necessary exist a monochromatic congruent copy of $T$? Here "monochromatic" means that all three vertices receive the same color. It is known that the answer depends on $T$.

There are several instances known where the answer is "yes". For example, an amusing exercise is to show that this holds if $T$ is a triangle with side lengths $1, \sqrt{3}, 2$. I believe Erdős and Graham gave infinite families of $T$ for which the answer is yes.

On the other hand, one can give a two-coloring of the plane so that there is no monochromatic triangle with side lengths $1, 1, 1$.

If I remember correctly, Erdős conjectured that there is always a monochromatic copy of $T$, except for the equilateral triangle which is the only exception.

(1) Does anyone know if there has been any recent progress on this conjecture?

(2) What I'd really like to know: what about the (degenerate) special case of a $1, 1, 2$ triangle? This question can be seen as a hypergraph analogue of the Hadwiger-Nelson problem, and suggests an interesting intersection of Euclidean Ramsey theory and additive combinatorics.