Using almost disjoint coding we prove the consistency of the existence of a definable ω-mad family of infinite subsets of ω together with.

We extend the work of Fischer et al. [6] by presenting a method for controlling cardinal characteristics in the presence of a projective wellorder and 2ℵ0>ℵ2. This also answers a question of Harrington [9] by showing that the existence of a Δ31 wellorder of the reals is consistent with Martinʼs axiom and 2ℵ0=ℵ3.

We show that is consistent with the existence of a -definable wellorder of the reals and a -definable ω-mad subfamily of [ω]ω.

It is relatively consistent with ZFC that every countable $FU_{fin} $ space of weight N₁ is metrizable. This provides a partial answer to a question of G. Gruenhage and P. Szeptycki [GS1].

If F ⊆ NN is an analytic family of pairwise eventually different functions then the following strong maximality condition fails: For any countable H ⊆ NN. no member of which is covered by finitely many functions from F, there is f ∈ F such that for all h ∈ H there are infinitely many integers k such that f(k) = h(k). However if V = L then there exists a coanalytic family of pairwise eventually different functions satisfying this strong maximality (...) condition. (shrink)

Using countable support iterations of S-proper posets, we show that the existence of a definable wellorder of the reals is consistent with each of the following: , and.

If dominating functions in ω ω are adjoined repeatedly over a model of GCH via a finite-support c.c.c. iteration, then in the resulting generic extension there are no long towers, every well-ordered unbounded family of increasing functions is a scale, and the splitting number s (and hence the distributivity number h) remains at ω 1.