Cardinal Coefficients of Tree Ideals, Antichain Numbers, and a link to Continuous Ramsey Theory

• Applicant: Professor Dr. Otmar Spinas
• Subject Area: Mathematics
• Term: since 2020
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 447849607

We investigate open problems about cardinal coefficients of classical tree ideals, especially their additivities and cofinalities. Such ideals are defined on the Cantor space, on the Baire space resp., by means of certain tree forcings consisting of subtrees of the full binary tree, the tree of all finite sequences of integers resp. A focal point of our research is the Marcewski ideal and the problem whether its additivity is less or equal the splitting number s or its countable version s_\sigma which is trivially larger than s. This question is linked to the old open problem whether s=s_\sigma.As usually the exact value of some cardinal invariant cannot be decided in ZFC, we investigate ZFC-models, i.e. models of the mathematical universe. By constructing a ZFC-model for a certain relation between some cardinal invariants the consistency of this relation with ZFC is shown. The most powerful method for constructing such models is forcing.By the definition of a tree ideal it is clear that for its investigation we need to understand the structure and size of maximal antichains of its associated tree forcing, in particular its antichain number, i.e. the least size of a nontrivial maximal antichain. A technical question that is of importance here is when the maximality of some antichain in some model is preserved an a forcing extension. Recent results of the applicant exhibit a link between the above problem whether the additivity of the Marcewski is below s_\sigma and the homogeneity number hm that comes up in the theory of continuous colorings of the Cantor plane. More precisely, if hm equals the size of the continuum then this inequality is true. It is known that in case hm is smaller than the continuum, then hm is its cardinal predecessor, hence the continuum is a successor cardinal. I guess that this is good evidence for the truth of our conjecture, as it seems strange that it depends on whether the continuum is a limit cardinal or not.Besides the Marcewski ideal there are several other tree ideals, some of which have already been studied quite thoroughly. Of particular interest to me is the splitting ideal that is defined on the Cantor space by its associated splitting tree forcing. No nontrivial upper bound for its additivity is known. Interesting candidates for this are the bounding number b and the covering coefficient of the meager ideal cov(M).One can also ask for the relation between the additivities of different tree ideals. One open problem here is whether the additivity of the Silver ideal is probvably below that of Marcewski's.By the theory of Tukey relations between partial orders there exists a certain duality between additivities and cofinalities. It implies that often a provable inequality between the additivities of two ideals implies the converse inequality between their cofinalities. Hence also cofinalities of tree ideals are a central topic of this proposal.

• DFG Programme: Research Grants