  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.
