Compactness of maximal eventually different families

We show that there is an effectively closed maximal eventually different family of functions in spaces of the form $\prod_n F(n)$ for $F\colon \mathbb{N} \to \mathbb{N}\cup\{\mathbb{N}\}$ and give an exact criterion for when there exists an effectively compact such family.Comment: 9 pages. Some small errors correcte

On Horowitz and Shelah's Borel maximal eventually different family

We show there is a closed (in fact effectively closed, i.e., $\Pi^0_1$) eventually different family (working in ZF or less).Comment: 7 pages. Some small errors correcte

Easton supported Jensen coding and projective measure without projective Baire

We prove that it is consistent relative to a Mahlo cardinal that all sets of reals definable from countable sequences of ordinals are Lebesgue measurable, but at the same time, there is a $\Delta^1_3$ set without the Baire property. To this end, we introduce a notion of stratified forcing and stratified extension and prove an iteration theorem for these classes of forcings. Moreover we introduce a variant of Shelah's amalgamation technique that preserves stratification. The complexity of the set which provides a counterexample to the Baire property is optimal.Comment: 142 page

Definable maximal discrete sets in forcing extensions

Let $\mathcal R$ be a $\Sigma^1_1$ binary relation, and recall that a set $A$ is $\mathcal R$-discrete if no two elements of $A$ are related by $\mathcal R$. We show that in the Sacks and Miller forcing extensions of $L$ there is a $\Delta^1_2$ maximal $\mathcal{R}$-discrete set. We use this to answer in the negative the main question posed in [5] by showing that in the Sacks and Miller extensions there is a $\Pi^1_1$ maximal orthogonal family ("mof") of Borel probability measures on Cantor space. A similar result is also obtained for $\Pi^1_1$ mad families. By contrast, we show that if there is a Mathias real over $L$ then there are no $\Sigma^1_2$ mofs.Comment: 16 page

The Ramsey property implies no mad families

We show that if all collections of infinite subsets of $\N$ 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 $E_0$, 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.

Lebesgue's Density Theorem and definable selectors for ideals

We introduce a notion of density point and prove results analogous to Lebesgue's density theorem for various well-known ideals on Cantor space and Baire space. In fact, we isolate a class of ideals for which our results hold. In contrast to these results, we show that there is no reasonably definable selector that chooses representatives for the equivalence relation on the Borel sets of having countable symmetric difference. In other words, there is no notion of density which makes the ideal of countable sets satisfy an analogue to the density theorem. The proofs of the positive results use only elementary combinatorics of trees, while the negative results rely on forcing arguments.Comment: 28 pages; minor corrections and a new introductio

Lightface Σ21\Sigma^1_2-indescribable cardinals

$\Sigma^1_3$-absoluteness for ccc forcing means that for any ccc forcing $P$, ${H_{\omega_1}}^V \prec_{\Sigma_2}{H_{\omega_1}}^{V^P}$. "$\omega_1$ inaccessible to reals" means that for any real $r$, ${\omega_1}^{L[r]}<\omega_1$. To measure the exact consistency strength of "$\Sigma^1_3$-absoluteness for ccc forcing and $\omega_1$ is inaccessible to reals", we introduce a weak version of a weakly compact cardinal, namely, a (lightface) $\Sigma^1_2$-indescribable cardinal; $\kappa$ has this property exactly if it is inaccessible and $H_\kappa \prec_{\Sigma_2} H_{\kappa^+}$

Maximal almost disjoint families, determinacy, and forcing

We study the notion of $\mathcal J$-MAD families where $\mathcal J$ is a Borel ideal on $\omega$. We show that if $\mathcal J$ is an arbitrary $F_\sigma$ ideal, or is any finite or countably iterated Fubini product of $F_\sigma$ ideals, then there are no analytic infinite $\mathcal J$-MAD families, and assuming Projective Determinacy there are no infinite projective $\mathcal J$-MAD families; and under the full Axiom of Determinacy + $V=\mathbf{L}(\mathbb{R})$ there are no infinite $\mathcal J$-mad families. These results apply in particular when $\mathcal J$ is the ideal of finite sets $\mathrm{Fin}$, which corresponds to the classical notion of MAD families. The proofs combine ideas from invariant descriptive set theory and forcing.Comment: 40 page