We advance the theory of strong colorings over partitions and obtain positive and negative results at the level of א1 and at higher cardinals. Improving a strong coloring theorem due to Galvin [Gal80], we prove that the existence of a non-meager set of reals of cardinality א1 is equivalent to the higher dimensional version of an unbalanced negative partition relation due to Erdős, Hajnal and Milner [EHM66]. We then prove that these colorings cannot be strengthened to overcome countable partitions of [א1], even in the presence of both a Luzin set and a coherent Souslin tree. A correspondence between combinatorial properties of partitions and chain conditions of natural forcing notions for destroying strong colorings over them is uncovered and allows us to prove positive partition relations for א1 from weak forms of Martin’s Axiom, thereby answering two questions from [CKS20]. Positive partition relations for א2 and higher cardinals are similarly deduced from the Generalized Martin’s Axiom. Finally, we provide a group of pump-up theorems for strong colorings over partitions. Some of them solve more problems from [CKS20].
|Original language||English GB|
|Number of pages||21|
|State||Published - 16 Apr 2021|