Ramsey Property and Pathological Sets: Almost Disjointness, Independence and Other Maximal Objects

Abstract

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 $\mathcal{E}\mathcal{D}$, ${\mathcal{E}\mathcal{D}}_{\mathrm{fin}}$, and $\finalphaα$ 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.