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Prediction problems and ultrafilters on ω

100%

Taylor A.

Fundamenta Mathematicae

2012

tom 219

nr 2

111-117

We consider prediction problems in which each of a countably infinite set of agents tries to guess his own hat color based on the colors of the hats worn by the agents he can see, where who can see whom is specified by a graph V on ω. Our interest is in the case in which 𝓤 is an ultrafilter on the set of agents, and we seek conditions on 𝓤 and V ensuring the existence of a strategy such that the set of agents guessing correctly is of 𝓤-measure one. A natural necessary condition is the absence of a set of agents in 𝓤 for which no one in the set sees anyone else in the set. A natural sufficient condition is the existence of a set of 𝓤-measure one so that everyone in the set sees a set of agents of 𝓤-measure one. We ask two questions: (1) For which ultrafilters is the natural sufficient condition always necessary? (2) For which ultrafilters is the natural necessary condition always sufficient? We show that the answers are (1) p-point ultrafilters, and (2) Ramsey ultrafilters.

Minimal predictors in hat problems

63%

Hardin C., Taylor A.

Fundamenta Mathematicae

2010

tom 208

nr 3

273-285

We consider a combinatorial problem related to guessing the values of a function at various points based on its values at certain other points, often presented by way of a hat-problem metaphor: there are a number of players who will have colored hats placed on their heads, and they wish to guess the colors of their own hats. A visibility relation specifies who can see which hats. This paper focuses on the existence of minimal predictors: strategies guaranteeing at least one player guesses correctly, regardless of how the hats are colored. We first present some general results, in particular showing that transitive visibility relations admit a minimal predictor exactly when they contain an infinite chain, regardless of the number of colors. In the more interesting nontransitive case, we focus on a particular nontransitive relation on ω that is elementary, yet reveals unexpected phenomena not seen in the transitive case. For this relation, minimal predictors always exist for two colors but never for ℵ₂ colors. For ℵ₀ colors, the existence of minimal predictors is independent of ZFC plus a fixed value of the continuum, and turns out to be closely related to certain cardinal invariants involving meager sets of reals.

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2 Fundamenta Mathematicae

2 Taylor A.

1 Hardin C.

1 2012

1 2010

Wyniki wyszukiwania

Prediction problems and ultrafilters on ω

Minimal predictors in hat problems