Ramsey Property and Pathological Sets: Almost Disjointness, Independence and Other Maximal Objects
Jialiang He, Jintao Luo, Shuguo Zhang

Abstract
We show that under , if the Ramsey property holds for all sets in a good pointclass , then there is no MAD family in , proving a long-standing conjecture made by A.R.D.\ Mathias in 1977. This also holds for -MAD families with respect to analytic ideals including , , and for all countable ordinals . Under the same assumption, we show that if any one of the Baire property, Lebesgue measurability or Ramsey property holds for all sets in , then there is no maximal independent family in . Under the stronger assumption , we further prove that if the Ramsey property holds for all sets in , then contains no Vitali sets and thus no Hamel bases.
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