Sierpinski's onto mapping and partitions

Menachem Kojman, Assaf Rinot, Juris Steprans

Research output: Working paper/PreprintPreprint

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Abstract

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.
Original languageEnglish
StatePublished - 2021

Publication series

NameArxiv preprint

Keywords

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

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