## Abstract

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.

(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.

Original language | American English |
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State | Published - 2021 |

## Keywords

- Primary 03E02, Secondary 03E35, 03E17
- math.LO