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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.