## Decompositions of the plane and the size of the continuum

### Volume 203 / 2009

Fundamenta Mathematicae 203 (2009), 65-74
MSC: Primary 03E50; Secondary 03E05, 51M05.
DOI: 10.4064/fm203-1-6

#### Abstract

We consider a triple $\langle E_0,E_1,E_2\rangle$ of equivalence
relations on $\mathbb{R}^2$ and investigate the possibility of
decomposing the plane into three sets $\mathbb{R}^2=S_0 \cup S_1 \cup
S_2$ in such a way that each $S_i$ intersects each $E_i$-class in
finitely many points. Many results in the literature, starting
with a famous theorem of Sierpiński, show that for certain
triples the existence of such a decomposition is equivalent to the
continuum hypothesis. We give a characterization in ZFC of the
triples for which the decomposition exists. As an application we
show that the plane can be covered by three *sprays*
regardless of the size of the continuum, thus answering a question
of J. H. Schmerl.