Abstract
In this chapter, we again have ω as the set of agents, but we only consider one-way visibility on ω (in which each agent can see some subset of the higher-numbered agents). We first use the μ-predictor from Chapter 1 to completely characterize those transitive graphs adequate for minimal and optimal predictors. The result for optimal predictors is then extended to nontransitive graphs (in the countable case).
The case of minimal predictors for nontransitive graphs turns out to be quite complex. We explore a particular nontransitive visibility graph called the “parity relation” in which even-numbered agents (in ω) see higher-numbered odds and vice versa. It turns out here that whether of not there is a predictor ensuring at least one correct guess is very dependent on the number of colors: with finitely many colors, the answer is yes, with ℵ 2 colors, the answer is no, and with denumerably many colors the answer is independent of ZFC plus a fixed value of the continuum.
Section 4.6 deals with P-point ultrafilters and Ramsey ultrafilters on ω. Section 4.7 provides a finer analysis of the investigations begun in Section 4.6. The chapter concludes with a brief glimpse at the work done by Andreas Blass and others on evasion and prediction.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Bibliography
Bartoszyński, T.: Invariants of measure and category. In: Foreman, M., Kanamori, A. (eds.) Handbook of Set Theory, vol. 1, pp. 491–555. Springer, Dordrecht (2010)
Bartoszyński, T., Judah, H.: Set theory: on the Structure of the Real Line. A K Peters, Wellesley (1995)
Blass, A.: Combinatorial cardinal characteristics of the continuum. In: Foreman, M., Kanamori, A. (eds.) Handbook of Set Theory, vol. 1, pp. 395–489. Springer, Dordrecht (2010)
Baumgartner, J.E., Taylor, A.D.: Partition theorems and ultrafilters. Trans. Am. Math. Soc. 241, 283–309 (1978)
Blass, A.: Cardinal characteristics and the product of countably many infinite cyclic groups. J. Algebra 169, 512–540 (1994)
Blass, A.: Reductions between cardinal characteristics of the continuum. In: Bartoszyński, T., Scheepers, M. (eds.) Contemporary Mathematics 192: Set Theory: Annual Boise Extravaganza in Set Theory (BEST) Conference, pp. 31–49 (1996)
Brendle, J.: Evasion and prediction – the Specker phenomenon and Gross spaces. Forum Math. 7, 513–541 (1995)
Brendle, J.: Evasion and prediction III: constant prediction and dominating reals. J. Math. Soc. Jpn. 55(1), 101–115 (2003)
Brendle, J., Shelah, S.: Evasion and prediction II. J. Lond. Math. Soc. 53, 19–27 (1996)
Brendle, J., Shelah, S.: Evasion and prediction IV: strong forms of constant prediction. Arch. Math. Logic 42, 349–360 (2003)
Eda, K.: On a Boolean power of a torsion-free abelian group. J. Algebra 82, 84–93 (1983)
Hardin, C.S.: On transitive subrelations of binary relations. J. Symb. Logic 76, 1429–1440 (2010)
Hardin, C.S., Taylor, A.D.: Minimal predictors in hat problems. Fundam. Math. 208, 273–285 (2010)
Jech, T.J.: Set Theory: The Third Millenium Edition, Revised and Expanded. Springer Monographs in Mathematics. Springer, Berlin (2003)
Kada, M.: The Baire category theorem and the evasion number. Proc. Am. Math. Soc. 126(11), 3381–3383 (1998)
Kamo, S.: Cardinal invariants associated with predictors. In: Buss, S. et al. (eds.) Logic Colloquium’98, Prague. Lecture Notes in Logic, vol. 13, pp. 280–295 (2000)
Kamo, S.: Cardinal invariants associated with predictors II. J. Math. Soc. Jpn. 53, 35–57 (2001)
Miller, A.W.: Some properties of measure and category. Trans. Am. Math. Soc. 266(1), 93–114 (1981)
Specker, E.: Additive Gruppen von Folgen ganzer Zahler. Port. Math. 9, 131–140 (1950)
Taylor, A.D.: Prediction problems and ultrafilters on ω. Fundam. Math. 219, 111–117 (2012)
Velleman, D.J.: The even-odd hat problem (2011, preprint)
Wimmers, E.: The Shelah P-point independence theorem. Isr. J. Math. 4, 28–48 (1982)
Yiparaki, O.: On some tree partitions. PhD thesis, University of Michigan (1994)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Hardin, C.S., Taylor, A.D. (2013). The Denumerable Setting: One-Way Visibility. In: The Mathematics of Coordinated Inference. Developments in Mathematics, vol 33. Springer, Cham. https://doi.org/10.1007/978-3-319-01333-6_4
Download citation
DOI: https://doi.org/10.1007/978-3-319-01333-6_4
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-01332-9
Online ISBN: 978-3-319-01333-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)