Some partitions of three-dimensional combinatorial cubes (Q1337176)

The author investigates various partition relations of three-dimensional cubes. So he shows in Section 2 that \(\aleph_ 2 \to [\aleph_ 1]^ 3_{\aleph_ 1}\) is equivalent to Chang's conjecture. On the other hand, \(\aleph_ 2 \to [\aleph_ 1]^ 2_{\aleph_ 1}\) is a consequence of CH and not equivalent to Chang's conjecture. As main result of the paper, the author shows that \(\aleph_ 2 \nrightarrow [\aleph_ 1]^ 3_{\aleph_ 0}\) without additional set-theoretic assumptions. This shows that under Galvin's conjecture the continuum must be much bigger than \(\aleph_ 2\). Further, the author investigates continuous partitions of \([A]^ r\) where \(A\) is a set of reals and the topology is the natural one. For instance, he constructs a continuous partition \(c:[\mathbb{Q}]^ 3 \to \mathbb{Q}\) such that \(c''[X]^ 3 = \mathbb{Q}\) for every \(X \subseteq \mathbb{Q}\) which is homeomorphic to \(\mathbb{Q}\). He also constructs an uncountable \(X \subseteq \mathbb{R}\) and a continuous \(c : [X]^ 3 \to X\) such that \(c''[Y]^ 3 = X\) for every uncountable \(Y \subseteq X\). The paper ends with some remarks and open problems.