A Sacks indestructible co-analytic maximal eventually different family

A Co-analytic Cohen-Indestructible Maximal cofinitary Group

Assuming that every set is constructible, we find a Π1 maximal cofinitary group of permutations of N which is indestructible by Cohen forcing. Thus we show that the existence of such groups is consistent with arbitrarily large continuum. Our method also gives a new proof, inspired by the forcing method, of Kastermans’ result that there exists a Π1 maximal cofinitary group in L.

A co-analytic maximal set of orthogonal measures

We prove that if V = L then there is a Π11 maximal orthogonal (i.e. mutually singular) set of measures on Cantor space. This provides a natural counterpoint to the well-known Theorem of Preiss and Rataj [16] that no analytic set of measures can be maximal orthogonal.

MAD families and the rationals

Rational numbers are used to classify maximal almost disjoint (MAD) families of subsets of the integers. Combinatorial characterization of indestructibility of MAD families by the likes of Cohen, Miller and Sacks forcings are presented. Using these it is shown that Sacks indestructible MAD family exists in ZFC and that b = c implies that there is a Cohen indestructible MAD family. It follows th...

Eventually Different Functions and Inaccessible Cardinals

Forcing indestructibility of MAD families

Let A ⊆ [ω]ω be a maximal almost disjoint family and assume P is a forcing notion. Say A is P-indestructible if A is still maximal in any P-generic extension. We investigate P-indestructibility for several classical forcing notions P. In particular, we provide a combinatorial characterization of P-indestructibility and, assuming a fragment of MA, we construct maximal almost disjoint families wh...