Ramsey, Lebesgue, and Marczewski sets and the Baire property

We investigate the completely Ramsey, Lebesgue, and Marczewski a-algebras and their relations to the Baire property in the Ellentuck and density topologies. Two theorems concerning the Marczewski a-algebra (s) are presented. THEOREM. In the density topology D, (s) coincides with the sigma-algebra of Lebesgue measurable sets. THEOREM. In the Ellentuck topology on [omega](omega), (s)(0) is a proper subset of the hereditary ideal associated with (s). We construct an example in the Ellentuck topology of a set which is first category and measure 0 but which is not B-r-measurable. In addition, several theorems concerning perfect sets in the Ellentuck topology are presented. In particular, it is shown that there exist countable perfect sets in the Ellentuck topology.