Coanalytic ultrafilter bases

We study the definability of ultrafilter bases on \(\omega \) in the sense of descriptive set theory. As a main result we show that there is no coanalytic base for a Ramsey ultrafilter, while in L we can construct \(\Pi ^1_1\) P-point and Q-point bases. We also show that the existence of a \({\varvec{\Delta }}^1_{n+1}\) ultrafilter is equivalent to that of a \({\varvec{\Pi }}^1_n\) ultrafilter base, for \(n \in \omega \). Moreover we introduce a Borel version of the classical ultrafilter number and make some observations.