We show that it is consistent with ZFC that the family of functions with the Baire property has the difference property. That
is, every function for which f(x + h)-f(x) has the Baire property for every h∈R is of the form f=g + Awhere g has the Baire property and A is additive.
We consider Problems 2 and 3 in  asked by M. Laczkovich concerning the difference property of Borel measurable functions.
We show that the axiom of determinacy implies affirmative answer to Problem 2 (Theorem 2) and that Problem 3 is settled affirmatively
for all infinite order Baire classes (Theorem 1.)
Authors:Krzysztof Ciesielski and Janusz Pawlikowski
We prove that axiom CPAgameprism, which follows from the Covering Property Axiom CPA and holds in the iterated perfect set model, implies that there exists
a Hamel basis which is a union of less than continuum many pairwise disjoint perfect sets. We will also give two consequences
of this last fact.