Let $\omega$ be the first infinite ordinal and let $\omega_1$ be the first uncountable ordinal. Citing results of Todorcevic and Hajnal, Jones says that if you color pairs of elements of $\omega_1$ in blue and red, then it is independent of ZFC to decide whether there must be either a blue subset of type $\omega_1$ or a red subset of type $\omega+2$.