SOLVED

Is it true that in any finite colouring of the integers there exists a monochromatic solution to
\[\frac{1}{a}=\frac{1}{b}+\frac{1}{c}\]
with distinct $a,b,c$?

The density version of this is

[302]. This colouring version is true, as proved by Brown and Rödl

[BrRo91].