Definable maximal cofinitary groups of intermediate size

Using almost disjoint coding, we show that for each $1<M<N<\omega$ consistently $\mathfrak{d}=\mathfrak{a}_g=\aleph_M<\mathfrak{c}=\aleph_N$, where $\mathfrak{a}_g=\aleph_M$ is witnessed by a $\Pi^1_2$ maximal cofinitary group.Comment: 22 page

Projective maximal families of orthogonal measures with large continuum

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

Note on s0s_0 nonmeasurable unions

In this note we consider an arbitrary families of sets of $s_0$ ideal introduced by Marczewski-Szpilrajn. We show that in any uncountable Polish space $X$ and under some combinatorial and set theoretical assumptions (cov(s_0)=\c for example), that for any family \ca\subseteq s_0 with \bigcup\ca =X, we can find a some subfamily \ca'\subseteq\ca such that the union \bigcup\ca' is not $s$-measurable. We have shown a consistency of the cov(s_0)=\omega_1<\c and existence a partition of the size $\omega_1$ \ca\in [s_0]^{\omega} of the real line \bbr, such that there exists a subfamily \ca'\subseteq\ca for which \bigcup\ca' is $s$-nonmeasurable. We also showed that it is relatively consistent with ZFC theory that \omega_1<\c and existence of m.a.d. family \ca such that \bigcup\ca is $s$-nonmeasurable in Cantor space $2^\omega$ or Baire space $\omega^\omega$. The consistency of $a<cov(s_0)$ and $cov(s_0)<a$ is proved also.Comment: 12 page