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    Projective maximal families of orthogonal measures with large continuum

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    We study maximal orthogonal families of Borel probability measures on 2ω2^\omega (abbreviated m.o. families) and show that there are generic extensions of the constructible universe LL in which each of the following holds: (1) There is a Δ31\Delta^1_3-definable well order of the reals, there is a Π21\Pi^1_2-definable m.o. family, there are no Σ21\mathbf{\Sigma}^1_2-definable m.o. families and b=c=ω3\mathfrak{b}=\mathfrak{c}=\omega_3 (in fact any reasonable value of c\mathfrak{c} will do). (2) There is a Δ31\Delta^1_3-definable well order of the reals, there is a Π21\Pi^1_2-definable m.o. family, there are no Σ21\mathbf{\Sigma}^1_2-definable m.o. families, b=ω1\mathfrak{b}=\omega_1 and c=ω2\mathfrak{c}=\omega_2.Comment: 12 page

    The Ramsey property implies no mad families

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    We show that if all collections of infinite subsets of N\N have the Ramsey property, then there are no infinite maximal almost disjoint (mad) families. This solves a long-standing problem going back to Mathias \cite{mathias}. The proof exploits an idea which has its natural roots in ergodic theory, topological dynamics, and invariant descriptive set theory: We use that a certain function associated to a purported mad family is invariant under the equivalence relation E0E_0, and thus is constant on a "large" set. Furthermore we announce a number of additional results about mad families relative to more complicated Borel ideals.Comment: 10 pages; fixed a mistake in Theorem 4.
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