Prediction problems and ultrafilters on $\omega$

Alan D. Taylor¹

¹ Department of Mathematics Union College Schenectady, NY 12308, U.S.A.

Fundamenta Mathematicae, Tome 219 (2012) no. 2, pp. 111-117.

Résumé

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 $\omega$. Our interest is in the case in which $\mathcal{U}$ is an ultrafilter on the set of agents, and we seek conditions on $\mathcal{U}$ and $V$ ensuring the existence of a strategy such that the set of agents guessing correctly is of $\mathcal{U}$-measure one. A natural necessary condition is the absence of a set of agents in $\mathcal{U}$ for which no one in the set sees anyone else in the set. A natural sufficient condition is the existence of a set of $\mathcal{U}$-measure one so that everyone in the set sees a set of agents of $\mathcal{U}$-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.

Détail

DOI : 10.4064/fm219-2-3

Keywords: consider prediction problems which each countably infinite set agents tries guess his own hat color based colors hats worn agents see where who see whom specified graph omega interest which mathcal ultrafilter set agents seek conditions mathcal ensuring existence strategy set agents guessing correctly mathcal measure natural necessary condition absence set agents mathcal which set sees anyone else set natural sufficient condition existence set mathcal measure everyone set sees set agents mathcal measure ask questions which ultrafilters natural sufficient condition always necessary which ultrafilters natural necessary condition always sufficient answers p point ultrafilters ramsey ultrafilters