Sierpiński proved that a version of Ramsey's theorem for colourings of pairs of countable ordinals fails miserably by comparing the ordering of $\omega_1$ with the linear ordering of (a subset of) the real line. This counterexample is thus reliant on some form of AC, hence quite non-constructive.

Are there any positive results concerning Ramsey's theorem for $\omega_1$ that would assume some regularity conditions on the colouring? (For example, such as being Borel with respect to the order topology.)