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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.

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

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