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Roughly speaking, SOCA (Semi Open Coloring Axiom) says that for an open coloring of the unordered pairs over an uncountable separable metric space you can always find an uncountable homogeneous subset (you can find a more detail description of SOCA in Kunen's Set Theory book).

Here uncountable only means something of size at least $\aleph_{1}$. This makes us think that you can increase that, for example, to something of size at least $\aleph_{2}$ (name this axiom SOCA(\aleph_{2})).

Has someone work on this? What it is known? Is it consistent?

My understanding is that SOCA is equiconsistent with ZFC and has the same flavor as OCA (but weaker). Furthermore, Shelah poved that SOCA is compatible with $2^{\aleph_{0}}>\aleph_{2}$ so this question makes sense. Also, the existence of entangled sets of size continumm let us show that SOCA($\aleph_{2}$) implies that $2^{\aleph_{0}}>\aleph_{2}$.

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    $\begingroup$ I think SOCA + SOCA($\aleph_2$) implies $\mathfrak{b} > \aleph_2$. It's not clear that SOCA($\aleph_2$) is enough by itself, however, since it only applies to spaces of size $\aleph_2$. $\endgroup$
    – Paul McKenney
    Commented Mar 30, 2015 at 22:02
  • $\begingroup$ I found out that this question was stated in Shelah's article 153 " On the consistency of some partition theorems for continuous colorings, and the structure of $\aleph_ 1$-dense real order types ". This is an article of Shelah, Rubin and Abraham. It appears as question 1.11. $\endgroup$
    – Iván Ongay Valverde
    Commented Apr 14, 2015 at 2:22

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