Partition properties for simply definable colourings

TL;DR

This paper investigates how large cardinal assumptions and forcing axioms influence partition properties of definable colourings on uncountable regular cardinals, revealing differences between  and  and exploring implications for inaccessible cardinals.

Contribution

It establishes new connections between large cardinal properties, forcing axioms, and partition properties for  and , especially for -definable colourings, and analyzes their consistency strengths.

Findings

Large cardinal assumptions imply homogeneous closed unbounded subsets for -definable colourings.

Certain large cardinals lead to strong partition properties at inaccessible cardinals.

Martin's Maximum is compatible with -colourings lacking uncountable homogeneous sets.

Abstract

We study partition properties for uncountable regular cardinals that arise by restricting partition properties defining large cardinal notions to classes of simply definable colourings. We show that both large cardinal assumptions and forcing axioms imply that there is a homogeneous closed unbounded subset of $\omega_1$ for every colouring of the finite sets of countable ordinals that is definable by a $\Sigma_1$-formula that only uses the cardinal $\omega_1$ and real numbers as parameters. Moreover, it is shown that certain large cardinal properties cause analogous partition properties to hold at the given large cardinal and these implications yield natural examples of inaccessible cardinals that possess strong partition properties for $\Sigma_1$-definable colourings and are not weakly compact. In contrast, we show that $\Sigma_1$-definability behaves fundamentally different at $\omega_2$ by showing that various large cardinal assumptions and \emph{Martin's Maximum} are compatible with the existence of a colouring of pairs of elements of $\omega_2$ that is definable by a $\Sigma_1$-formula with parameter $\omega_2$ and has no uncountable homogeneous set. Our results will also allow us to derive tight bounds for the consistency strengths of various partition properties for definable colourings. Finally, we use the developed theory to study the question whether certain homeomorphisms that witness failures of weak compactness at small cardinals can be simply definable.