Alessandro Andretta: How many sprays cover the space?

Alessandro Andretta: How many sprays cover the space?

Abstract:
A spray in $ \mathbb{R}^d $ is a subset with a point called center, such
that every sphere with that center intersects the spray in a finite set.
J. Schmerl introduced sprays when $ d = 2$, and he proved that $
2^{\aleph_0 }  \leq \aleph_n $ iff the plane can be covered with $ n + 2
$ sprays with collinear centers.
We generalize this to all dimensions $d \geq 3$. In particular we show
that $ 2^{\aleph_0 } \leq \aleph_n $ iff $ \mathbb{R}^3 $ can be covered
with $ 2 n + 3 $ sprays such that their centers are  coplanar and no
three of them are collinear.
This is joint work with Ivan Izmestiev.