We study polychromatic Ramsey theory with a focus on colourings of [ω 2]2. We show that in the absence of GCH there is a wide range of possibilities. In particular each of the following is consistent relative to the consistency of ZFC: (1) 2ω = ω 2 and ${\omega }_2{\to }^{poly}{\left(\alpha \right)}_{{\aleph }_0-bdd}^2$ for every α <ω 2; (2) 2ω = ω 2 and ${\omega }_2{\nrightarrow }^{poly}{\left({\omega }_1\right)}_{2-bdd}^2$ .