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May I ask where can I read the example of a q-point which is not Ramsey. I'm especially looking for a coloring of the set of two element subsets of $\mathbb{N}$ without a homogeneous set in the q-point.

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Assume CH (or MA, or some other hypothesis that gives you a good supply of Ramsey ultrafilters). Let $U,V_0,V_1,\dots$ be non-isomorphic Ramsey ultrafilters on the set $\omega$ of natural numbers. Define $W$ to be the collection of those $X\subseteq\omega\times\omega$ such that, for $U$-almost all $m\in\omega$, the "slice" $\{n\in\omega:(m,n)\in X\}$ is in $V_m$. This ultrafilter $W$ is a Q-point but not Ramsey. (If you want an ultrafilter on $\omega$, move $W$ there by your favorite bijection.) The projection of $\omega^2$ to the first factor is neither one-to-one nor constant on any set in $W$, so $W$ isn't Ramsey. The proof that it's a Q-point takes a bit longer, and I'm between lectures at a conference, so here's just the key ingredient: Any one-to-one images of the $V_n$, say $f_n(V_n)$, are not only distinct (because the $V_n$ are not isomorphic) but contain sets $A_n$ that are pairwise disjoint (because they're P-points).

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