The colouring existence theorem revisited (Q2330054)

A the main result of this paper is devoted to a proof of a statement, abbreviated as \(\mathbf{Pr}_1(\lambda,\lambda,\lambda,(\theta_0,\theta_1))\), for a successor \(\lambda\) of a regular cardinal with \(\lambda\ge\theta_1^+\) and \(\theta_1>\theta_0\ge\aleph_0\). The general statement \(\mathbf{Pr}_1(\lambda,\mu,\sigma,(\theta_0,\theta_1))\) asserts that there is a colouring \(c:[\lambda]^2\to\sigma\) with the property that whenever two matrices \(\langle \zeta(\varepsilon,\alpha,i):\alpha<\mu, i<\mathbf{i}_\varepsilon\rangle\) (\(\varepsilon=0,1\)) of ordinals in \(\lambda\), with no repeated entries and with disjoint ranges, and \(\gamma<\sigma\) are given there are \(\alpha_0<\alpha_1<\mu\) such that \(c\{\zeta(0,\alpha_0,i),\zeta(1,\alpha_1,j)\}=\gamma\) for all \(i<\mathbf{i}_0\) and \(j<\mathbf{i}_1\).\par The author also discusses some variants of this statement and their relations to earlier work on negative partition relations. There are also applications in general topology relating to the existence of regular (zero-dimensional) spaces with various compactness properties and discretely untouchable points.