The Denumerable Setting: Full Visibility

  • Christopher S. Hardin
  • Alan D. Taylor
Part of the Developments in Mathematics book series (DEVM, volume 33)


With two colors, it is easy to produce a predictor for a denumerably infinite set of agents ensuring infinitely many will guess correctly. For example, even if agents only see higher-numbered agents, one can have an agent guess red if he sees infinitely many red hats, and guess green otherwise. If there are infinitely many red hats, everyone will guess red and the agents with red hats will be correct; if there are finitely many red hats, everyone will guess green, and the cofinitely many agents with green hats will be correct. But something much more striking is true: there is a predictor ensuring that all but finitely many—not just infinitely many—are correct, and this is what Gabay and O’Connor obtained using the axiom of choice.

This chapter presents that result, and uses it to derive a theorem of Hendrik Lenstra that exhibits a predictor ensuring that the guesses are either all correct or all incorrect. We also consider the kind of sequential guessing that arose in Chapter 2, and we prove an equivalence between this and Lenstra’s context. We also examine the necessity of the axiom of choice in these results, both via the property of Baire and via square bracket partition relations.


  1. [Gal65]
    Galvin, F.: Problem 5348. Am. Math. Mon. 72, 1136 (1965)MathSciNetGoogle Scholar
  2. [GP76]
    Galvin, F., Prikry, K.: Infinitary Jonsson algebras and partition relations. Algebra Univ. 6(3), 367–376 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  3. [HT08a]
    Hardin, C.S., Taylor, A.D.: An introduction to infinite hat problems. Math. Intell. 30(4), 20–25 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  4. [HT10]
    Hardin, C.S., Taylor, A.D.: Minimal predictors in hat problems. Fundam. Math. 208, 273–285 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  5. [JS93]
    Judah, H., Shelah, S.: Baire property and axiom of choice. Isr. J. Math. 84, 435–450 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  6. [Mat77]
    Mathias, A.R.D.: Happy families. Ann. Math. Logic 12, 59–111 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  7. [She84]
    Shelah, S.: Can you take Solovay’s inaccessible away? Isr. J. Math. 48, 1–47 (1984)CrossRefzbMATHGoogle Scholar
  8. [Sil66]
    Silverman, D.L.: Solution of problem 5348. Am. Math. Mon. 73, 1131–1132 (1966)CrossRefGoogle Scholar
  9. [Sol70]
    Solovay, R.: A model of set theory in which every set of reals is Lebesgue measurable. Ann. Math. 92, 1–56 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  10. [Tho67]
    Thorp, B.L.D.: Solution of problem 5348. Am. Math. Mon. 74, 730–731 (1967)MathSciNetCrossRefGoogle Scholar
  11. [Yip94]
    Yiparaki, O.: On some tree partitions. PhD thesis, University of Michigan (1994)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2013

Authors and Affiliations

  • Christopher S. Hardin
    • 1
  • Alan D. Taylor
    • 2
  1. 1.New YorkUSA
  2. 2.Department of MathematicsUnion CollegeSchenectadyUSA

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