Roghly speaking, SOCA (Semi Open Coloring Axiom) says that for an open coloring of an axioms 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 meake 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 existance of entangled sets of size continumm let us show that SOCA($\aleph_{2}$) implies that $2^{\aleph_{2}}>\aleph_{2}$.

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 $2^{\aleph_{0}}>\aleph_{2}$.