Partitioning the pairs and triples of topological spaces (Q915732)

This paper contains contributions to topological partition theory. The main results are: 1. The existence, for every uncountable \(\kappa\), of a space X such that \(X\to (B_{\kappa})^ 2_{\omega}\), where \(B_{\kappa}\) is \(\kappa +1\) with the limit ordinals below \(\kappa\) removed. In fact X can be taken to be a cardinal with the order topology and if \(\kappa\) is weakly compact then \(X=\kappa^+.\) 2. For every Hausdorff space X a partition of \([X]^ 2\) into two pieces without any dense-in-itself homogeneous set. 3. Negative results about \(X\to (Y)^ n_{\omega}:\) if for every \(\kappa\) of countable cofinality both \(\kappa^{\omega}=\kappa^+\) and \(\square_{\kappa}\) hold then for every regular X the relations \(X\to (top \omega +1)^ 2_{\omega}\) and \(X\to (Y)^ 3_{\omega}\) (Y countable and nondiscrete) are false.