A question which was mentioned in passing by Larson when discussing geometric set theory.
Are there models of set theory where all sets of reals have the Ramsey property but there is a set of reals without the Baire property?
A set $A \subseteq [\omega]^\omega$ has the Ramsey property iff it has an infinite homogeneous set, i.e. there is $H \subseteq \omega$ with either $[H]^\omega \subseteq A$ or $[H]^\omega \cap A = \emptyset$. (We use the term "reals" loosely to refer to $\omega^\omega$, $[\omega]^\omega$, and similar nonempty perfect Polish spaces.)
A set $A \subseteq \omega^\omega$ has the Baire property iff it is equal to an open set modulo the meagre ideal.
A possibly easier question is whether it is consistent that every set of reals is completely Ramsey and there is a set of reals without the Baire property. Equivalently, does the existence of a set of reals which does not have the Baire property with respect to the usual (Polish) topology imply the existence of a set of reals which does not have the Baire property with respect to the Ellentuck topology?
Here "completely Ramsey" means Ramsey strengthened so that the homogeneous set can be chosen below any given Mathias condition. It is well-known that a set of reals is completely Ramsey iff it has the Baire property with respect to the Ellentuck topology.
