Sierpinski's onto mapping and partitions

In his 1987 paper, Todorˇcevi´c remarks that Sierpi´nski’s onto mapping principle (1932) and the Erd˝os-Hajnal-Milner negative Ramsey relation
(1966) are equivalent to each other, and follow from the existence of a Luzin
set. Recently, Guzman and Miller showed that these two principles are also
equivalent to the existence of a nonmeager set of reals of cardinality ℵ1. We
expand this circle of equivalences and show that these propositions are equivalent also to the high-dimensional version of the Erd˝os-Hajnal-Milner negative Ramsey relation, thereby improving a CH theorem of Galvin (1980).
Then we consider the validity of these relations in the context of strong
colorings over partitions and prove the consistency of a positive Ramsey relation, as follows: It is consistent with the existence of both a Luzin set and of
a Souslin tree that for some countable partition p, all colorings are p-special.