A non-principal ultrafilter $\mathcal{U}$ on $\omega$ is a **p-point** (or **weakly selective**) iff for every partition $\omega = \bigsqcup \_{n < \omega} Z\_n$ into null sets, i.e each $Z\_n \not \in \mathcal{U}$, there exists a measure one set $S \in \mathcal{U}$ such that $S \cap Z\_n$ is finite for each $n$.


A non-principal ultrafilter $\mathcal{U}$ on $\omega$ is **Ramsey** (or **selective**) iff for every partition as above, there exists a measure one set $S$ such that $|S \cap Z\_n| = 1$ for each $n$.


**Clearly, every Ramsey ultrafilter is a p-point. What is known about the converse?**


I couldn't find anything, not even a consistency result, in any searches I've done or sources I've checked. Is very little known/published about the converse?



Is every p-point ultrafilter Ramsey?
Amit: 


The converse is not true, not even under MA. This is a result of Kunen, and the paper you want to look at is "Some points in $\beta{\mathbb N}$", Math. Proc. Cambridge Philos. Soc. 80 (1976), no. 3, 385–398. 


There is a related notion, called **$q$-point**. These are ultrafilters such that any finite-to-one $f:\omega\to\omega$ is injective on a set in the ultrafilter. A Ramsey ultrafilter is one that is simultaneously a $p$-point, and a $q$-point.


Miller proved ("There are no $Q$-points in Laver's model for the Borel conjecture", Proc. Amer. Math. Soc. 78 (1980), no. 1, 103–106) that it is consistent that there are no $q$-points. The consistency of the non-existence of $p$-points is significantly harder, and due to Shelah (see for example Chapter VI of his "Proper and improper forcing"). 


There is a fairly extensive literature on related results. You may want to start by looking at Blass' article in the Handbook of Set Theory, "Combinatorial Cardinal Characteristics of the Continuum". 


