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.)

  • $\begingroup$ I am not sure that the order topology could play a significant role here, as it is a very fine topology (for example, every set of successor ordinals is open, and the sets of successor ordinals < $\omega_1$ is homeomorphic to $\omega_1$; however, I don't manage to turn this into a proper counterexample). Topology plays a role in Galvin-Prikry's theorem because it's related to the idea of approximating a set with its initial segments, but it's not the same here... Maybe considering some form of definability could be better? However, I don't know which definability... $\endgroup$ – N. de Rancourt Feb 5 '18 at 15:31
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    $\begingroup$ In the model of ZF + the axiom of Determinacy, $\omega_1$ is measurable (in fact there are a lot stronger partition relations involving infinite exponents). Now suppose you live in a universe satisfying ZFC + the proper forcing axiom, and only consider colorings coming from those that live in $L(\mathbb{R})$ (constructible sets relative to the real numbers), then you will lots of positive results. $\endgroup$ – Jing Zhang Feb 5 '18 at 16:57

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