Dual Hat Problems, Ideals, and the Uncountable
We begin the chapter by studying “dual hat problems” where, roughly speaking, notions corresponding to injective functions or subsets are altered by considering surjective functions or partitions. Within the hat problem metaphor this shifts the focus from near-sightedness to colorblindness. It turns out that the μ-predictor has something to say in the dual hat problem as well.
We then move on to ideals on both countable and uncountable sets and investigate various ideal theoretic properties (weak P-pointedness, weak Q-pointedness, weak selectivity) and partition relations in this context. We also see the role played by non-regular ultrafilters as we try to extend an earlier result from the countable to the uncountable. Finally, we establish the equivalence of a hat-problem and the GCH.
KeywordsVisibility Graph Large Cardinal Transitive Graph Partition Relation Graph Versus
- [EHMR84]Erdős, P., Hajnal, A., Máté, A., Rado, R.: Combinatorial Set Theory: Partition Relations for Cardinals. North-Holland, Amsterdam (1984)Google Scholar
- [GR71]Graham, R., Rothschild, B.: A survey of finite Ramsey theory. In: Proceedings of the 2nd Louisiana Conference on Combinatorics, Graph Theory and Computing, Baton Rouge, pp. 21–40 (1971)Google Scholar
- [Yip94]Yiparaki, O.: On some tree partitions. PhD thesis, University of Michigan (1994)Google Scholar