The Mathematics of Coordinated Inference by Christopher S. Hardin & Alan D. Taylor

Author:Christopher S. Hardin & Alan D. Taylor
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

Dual hat problems represent an area largely unexplored. Nevertheless, we leave them now and turn to a consideration of ideals on uncountable cardinals and on ω.

5.3 Hat Problems and Ideals

In Sect. 4.​2 we considered the problem of characterizing those transitive visibility graphs on ω that yield a finite-error predictor and those that yield a minimal predictor, and Theorems 4.2.1 and 4.2.2 solved these problems. We begin this section by generalizing those results to the context of ideals on uncountable cardinals. As in Sect. 4.​2 we only consider one-way visibility in this section.

If I is an ideal on the infinite cardinal κ, then the notation denotes the assertion that for every set X ∈ I + and every function f : [X]2 → 2, there exists a set Y ⊆ X such that either Y ∈ I + and f([Y ]2) = 0 or |Y | =ω and f([Y ]2) = 1. Ramsey’s theorem asserts that when I = [ω]<ω , the Dushnik-Miller-Erdős theorem [EHMR84] asserts that when for any (infinite) cardinal κ, and it is well known that if κ is regular, then , where NS κ is the ideal of nonstationary subsets of κ.

Theorems 4.2.1 and 4.2.2 showed that if I = [ω]<ω and V is a transitive graph on ω, then there exists a positive I-measure predictor iff V contains an infinite complete subgraph, and there exists an I-measure one predictor iff V contains no infinite independent subgraph. The following generalizes this to transitive graphs and arbitrary ideals on an uncountable cardinal.

Theorem 5.3.1.

Suppose that I is an ideal on κ and V is an undirected transitive graph on κ. Consider the hat problem with one-way visibility given by V. Then (1) and (2) are equivalent, (3) implies (4), and, if , then (4) implies (3) and so they too are equivalent.

1. There exists a positive I-measure predictor for two colors.

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