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

理想树的基数系数、反链数以及与连续拉姆齐理论的联系

• 批准号: 447849607
• 负责人: Professor Dr. Otmar Spinas
• 金额: --
• 依托单位: Mathematisches Seminar
• 依托单位国家: 德国
• 项目类别: Research Grants
• 资助国家: 德国
• 项目状态: 未结题
• 来源: https://gepris.dfg.de/gepris/projekt/447849607?language=en
• 关键词: Cardinal; Coefficients; Tree; Ideals; Antichain

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.

本文研究了经典树理想的基数系数，特别是它们的可加性和共尾性。这样的理想分别定义在Cantor空间，Baire空间，通过由满二叉树的子树、所有有限整数序列的树和所有有限整数序列的树组成的某些树强迫，我们研究的一个重点是Marcewski理想及其可加性是否小于等于分裂数s或其可数形式s\sigma（比s平凡大）的问题。由于ZFC中某些基数不变量的精确值通常无法确定，我们研究了ZFC-模型，即数学论域的模型。通过对某些基数不变量之间的某种关系构造ZFC模型，证明了这种关系与ZFC的一致性。构造这类模型的最有力的方法是强迫，通过树理想的定义，很明显，为了研究它，我们需要了解它的树强迫的极大反链的结构和大小，特别是它的反链数，即非平凡极大反链的最小大小。这里有一个重要的技术问题是，当某个模型中的某个反链的极大性保持为强制扩张时。本申请人的最近结果展示了上述问题之间的联系，即Marcewski的可加性是否低于s_I sigma与康托平面的连续着色理论中出现的齐性数hm之间的联系。更准确地说，如果hm等于连续统的大小，那么这个不等式成立。已知如果hm小于连续统，则hm是它的基数前驱，因此连续统是后继基数。我想这是很好的证据，为真理的猜想，因为它似乎奇怪，这取决于是否连续是一个极限基数或没有。除了Marcewski理想有其他几个树理想，其中一些已经研究得相当彻底。我特别感兴趣的是在康托空间上定义的分裂理想，它是由其相关的分裂树强迫定义的。没有非平凡的上限，其可加性是已知的。有趣的候选者是边界数B和稀疏理想cov（M）的覆盖系数。人们还可以询问不同树理想的可加性之间的关系。这里的一个公开问题是银理想的可加性是否可能低于Marcewski理想的可加性。它意味着两个理想的可加性之间的可证明不等式往往意味着它们的共尾性之间的逆不等式。因此，树理想的共尾性也是这个建议的中心议题。