The Mathematics of Coordinated Inference pp 49-59 | Cite as

# Dual Hat Problems, Ideals, and the Uncountable

## Abstract

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.

## Keywords

Visibility Graph Large Cardinal Transitive Graph Partition Relation Graph Versus## Bibliography

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