  In his 1987 paper, Todorcevic remarks that Sierpinski's onto mapping
principle (1932) and the Erdos-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
$\aleph_1$. We expand this circle of equivalences and show that these
propositions are equivalent also to the high-dimensional version of the
Erdos-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.
