Ramsey Property and Pathological Sets: Almost Disjointness, Independence and Other Maximal Objects
We show that under $\mathsf{ZF} + \mathsf{CC}_{\mathbb R}$, if the Ramsey property holds for all sets in a good pointclass $\Gamma$, then there is no MAD family in $\Gamma$, proving a long-standing conjecture made by A.R.D.\ Mathias in 1977. This also holds for $\mathcal I$-MAD families with respect to analytic ideals $\mathcal I$ including $\mathcal{ED}$, $\mathcal{ED}_{\mathrm{fin}}$, and $\finalpha{\alpha}$ for all countable ordinals $\alpha$. Under the same assumption, we show that if any one of the Baire property, Lebesgue measurability or Ramsey property holds for all sets in $\Gamma$, then there is no maximal independent family in $\Gamma$. Under the stronger assumption $\mathsf{ZF} + \mathsf{DC}_{\mathbb R}$, we further prove that if the Ramsey property holds for all sets in $\Gamma$, then $\Gamma$ contains no Vitali sets and thus no Hamel bases.