We deal with the consistency strength of ZFC + variants of MA + suitable sets of reals are measurable (and/or Baire, and/or Ramsey). We improve the theorem of Harrington and Shelah [2] repairing the asymmetry between measure and category, obtaining also the same result for Ramsey. We then prove parallel theorems with weaker versions of Martin's axiom (\mathrm{MA}(\sigma-centered), (\mathrm{MA}(\sigma-linked)), \mathrm{MA}(\Gamma^+_{\aleph_0}), \mathrm{MA}(K)), getting Mahlo, inaccessible and weakly compact cardinals respectively. We prove that if there exists r \in \mathbf{R} such that \omega^{L\lbrack r\rbrack}_1 = \omega_1 and MA holds, then there exists a \triangle^1_3-selective filter on \omega, and from the consistency of ZFC we build a model for \mathrm{ZFC} + \mathrm{MA(I)} + every \triangle^1_3-set of reals is Lebesgue measurable, has the property of Baire and is Ramsey.