A classic result of Sierpiński shows that $2^{\aleph_0}\nrightarrow [\aleph_1]^2_2$, that is, there is a coloring of pairs of real numbers using two colors such that both colors appear on any uncountable set of reals.
Shelah proved [1] that the number of colors cannot be increased in general: starting with a measurable cardinal $\kappa$ in a model of GCH, there is a c.c.c. forcing $\mathbb{P}$ such that after forcing with $\mathbb{P}$, we have
- $2^{\aleph_0}=\kappa$, and
- $2^{\aleph_0}\rightarrow [\aleph_1]^2_3$,
which means that for any coloring $c$ of pairs of reals using three colors, we can find an uncountable $H$ on which $c$ assumes at most two colors.
The assumption of measurability can be weakened substantially, but I am curious if the conclusion implies that the continuum must be a weakly inaccessible (possible weakly Mahlo?) cardinal.
[1] Shelah, Saharon, Was Sierpinski right? I, Isr. J. Math. 62, No. 2-3, 355-380 (1988). ZBL0657.03028.
Edit: The notation used is a particular instance of “arrow notation” used to express various possible partition relations.
$\kappa\rightarrow [\theta]^\xi_\tau$ means
“if we color the $\xi$-tuples from $\kappa$ with $\tau$ colors, there is a set $H$ of cardinality $\theta$ on which the coloring omits at least one color.”
The “omits at least one color” is a very weak sort of homogeneity, and in the arrow notation it is indicated through the use of square brackets on the right side.
