# Does $2^{\aleph_0}\rightarrow [\aleph_1]^2_3$ require that the continuum is weakly inaccessible?

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.

• Would it be too far afield to define the $\to$ notation? Is there a difference between $2^{\aleph_0} \to …$ and $[2^{\aleph_0}] \to …$? Aug 5 at 17:44
• The first is a typo. I'll fix! Aug 5 at 18:25