Ramsey, Lebesgue, and Marczewski sets and the Baire property

We investigate the completely Ramsey, Lebesgue, and Marczewski σ-algebras and their relations to the Baire property in the Ellentuck and density topologies. Two theorems concerning the Marczewski σ-algebra (s) are presented.  THEOREM. In the density topology D, (s) coincides with the σ-algebra of Lebesgue measurable sets.  THEOREM. In the Ellentuck topology on ${\displaystyle {[\omega ]}^{\omega }}$, ${\displaystyle {\left(s\right)}_0}$ 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 ${\displaystyle B_r}$-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.