Partition properties for simply definable colourings (Q2182050)

Partition relations and strong colorings are central topics in infinitary combinatorics. This paper studies the possible partition properties that can hold, mostly at \(\omega_1\) and \(\omega_2\), for definable colorings. In this paper, definability is taken over \(V\). This differs from the work in [\textit{R. Bosch}, in: Set theory. Centre de Recerca Matemàtica Barcelona, 2003--2004. Basel: Birkhäuser. 55--82 (2006; Zbl 1113.03048)]. The main interest of this paper is the consistency of Ramsey-like properties for definable colorings over uncountable cardinals, in particular under the existence of large cardinals or forcing axioms. The paper starts by motivating and framing the problem. First, the author shows that for \(\Delta_0\)-colorings, ZFC proves a strong partition property for every regular uncountable cardinal. On the other hand, thepartition property for \(\Sigma_2\)-colorings with parameter \(z\) is equivalent to the same property for \(\Sigma_n\)-colorings, \(n \geq 2\), and in turn can be characterized using the existence of homogeneous sets for colorings from \(\mathrm{HOD}_z\). Since the existence of large cardinals as well as forcing axioms are consistent with \(V=\mathrm{HOD}\), it means that they cannot imply this partition property for \(\Sigma_2\)-colorings for any non-weakly compact cardinal. Focusing in \(\omega_1\) and \(\mathbf{\Sigma}_1\)-colorings, the author proves that many natural hypotheses, such as the existence of large cardinals or forcing axioms, imply that any \(\mathbf{\Sigma}_1\)-definable coloring on \(\omega_1\) has a homogeneous club. The author proves that consistency strength of this assertion is the existence of a measurable cardinal. Turing to \(\omega_2\), the author shows that the situation is quite different, and actually some forcing axioms outright imply the failure of the boldface \(\mathbf{\Sigma}_1\)-partition property for \(\omega_2\). Consistently, there is a parameter free counterexample, as consistently there is a parameter free definable well ordering of \(H(\omega_2)\), see [\textit{D. Asperó}, Ann. Pure Appl. Logic 146, No. 2--3, 150--179 (2007; Zbl 1116.03044); \textit{P. B. Larson}, Ann. Pure Appl. Logic 156, No. 1, 110--122 (2008; Zbl 1153.03035)]. It is still open whether \(\mathrm{MM}^{++}\) implies the partition property for \(\Sigma_1\)-colorings (without parameters) for \(\omega_2\). Besides \(\omega_1\) and \(\omega_2\), the paper contains a lot of information about the possibilities for various definable partition properties for other cardinals. In those cases, less is known and the paper includes some interesting open problems.