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Happy families and completely Ramsey sets

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Summary

We use games of Kastanas to obtain a new characterization of the classC ℱ of all sets that are completely Ramsey with respect to a given happy family ℱ. We then combine this with ideas of Plewik to give a uniform proof of various results of Ellentuck, Louveau, Mathias and Milliken concerning the extent ofC ℱ. We also study some cardinals that can be associated with the ideal ℐℱ of nowhere ℱ-Ramsey sets.

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References

  1. Balcar, B., Pelant, J., Simon, P.: The space of ultrafilters onN covered by nowhere dense sets. Fundam. Math.110, 11–24 (1980)

    Google Scholar 

  2. Balcar, B., Simon, P.: Disjoint refinement. In: Monk, J.D., Bonnet, R. (eds.): Handbook of Boolean algebras, vol. 2, pp. 333–388. Amsterdam: North-Holland 1989

    Google Scholar 

  3. Baumgartner, J.E., Laver, R.: Iterated perfect-set forcing. Ann. Math. Logic17, 271–288 (1979)

    Google Scholar 

  4. Blass, A.: Cardinal invariants, old and middle-aged. Lecture given in Oberwolfach, 1989

  5. van Douwen, E.K.: The integers and topology. In: Kunen, K., Vaughan, J.E. (eds.) Handbook of set-theoretic topology, pp. 111–167. Amsterdam: North-Holland 1984

    Google Scholar 

  6. Ellentuck, E.: A new proof that analytic sets are Ramsey. J. Symb. Logic39, 163–165 (1974)

    Google Scholar 

  7. Erdös, P., Rado, R.: Combinatorial theorems on classification of subsets of a given set. Proc. Lond. Math. Soc.2, 417–439 (1952)

    Google Scholar 

  8. Gale, D., Stewart, F.M.: Infinite games with perfect information. Ann. Math. Stud.28, 245–266 (1953)

    Google Scholar 

  9. Galvin, F.: A generalization of Ramsey's theorem. Notices Am. Math. Soc.15, 548 (1968)

    Google Scholar 

  10. Galvin, F., Prikry, K.: Borel sets and Ramsey's theorem. J. Symb. Logic38, 193–198 (1973)

    Google Scholar 

  11. Grigorieff, S.: Combinatorics on ideals and forcing. Ann. Math. Logic3, 363–394 (1971)

    Google Scholar 

  12. Hindman, N.: Finite sums from sequences within cells of a partition ofN. J. Comb. Theory Ser. A17, 1–11 (1974)

    Google Scholar 

  13. Judah, H., Miller, A.W., Shelah, S.: Sacks forcing Laver forcing and Martin's axiom. Arch. Math. Logic31, 145–161 (1992)

    Google Scholar 

  14. Kastanas, I.G.: On the Ramsey property for sets of reals. J. Symb. Logic48, 1035–1045 (1983)

    Google Scholar 

  15. Louveau, A.: Démonstration topologique des théorèmes de Silver et Mathias. Bull. Sci. Math., II. Ser.98, 97–102 (1974)

    Google Scholar 

  16. Louveau, A.: Une méthode topologique pour l'étude de la propriété de Ramsey. Isr. J. Math.23, 97–116 (1976)

    Google Scholar 

  17. Mansfield, R.: A footnote to a theorem of Solovay on recursive encodability. In: Macintyre, A., Pacholski, L., Paris, J. (eds.) Logic Colloquium '77, pp. 195–198. Amsterdam: North-Holland 1978

    Google Scholar 

  18. Matet, P.: Some filters of partitions. J. Symb. Logic53, 540–553 (1988)

    Google Scholar 

  19. Mathias, A.R.D.: Happy families. Ann. Math. Logic12, 59–111 (1977)

    Google Scholar 

  20. Milliken, K.R.: Completely separable families and Ramsey's theorem. J. Comb. Theory, Ser. A19, 318–334 (1975)

    Google Scholar 

  21. Nash-Williams, C.St.J.A.: On well-quasi-ordering transfinite sequences. Proc. Cambr. Phil. Soc.61, 33–39 (1965)

    Google Scholar 

  22. Pawlikowski, J.: Parametrized Ellentuck theorem. Topology Appl.37, 65–73 (1990)

    Google Scholar 

  23. Plewik, S.: On completely Ramsey sets. Fundam. Math.127, 127–132 (1986)

    Google Scholar 

  24. Plewik, S.: Isomorphic ideals (preprint)

  25. Prikry, K.: Determinateness and partitions. Proc. Am. Math. Soc.54, 303–306 (1976)

    Google Scholar 

  26. Ramsey, F.P.: On a problem of formal logic. Proc. Lond. Math. Soc.30, 264–286 (1929)

    Google Scholar 

  27. Repický, M.: Properties of measure and category in generalized Cohen's and Silver's forcing. Acta Univ. Carol., Math. Phys.28, 101–115 (1987)

    Google Scholar 

  28. Repický, M.: Collapsing of cardinals in generalized Cohen's forcing. Acta Univ. Carol., Math. Phys.29, 67–74 (1988)

    Google Scholar 

  29. Rothberger, F.: On some problems of Hausdorff and of Sierpiński. Fund. Math.35, 29–46 (1948)

    Google Scholar 

  30. Shelah, S.: On cardimal invariants of the continuum. Contemp. Math.31, 183–207 (1984)

    Google Scholar 

  31. Silver, J.: Every analytic set is Ramsey. J. Symb. Logic35, 60–64 (1970)

    Google Scholar 

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Authors and Affiliations

  1. Département de Mathématiques, Université de Caen, F-14032, Caen Cedex, France

    Pierre Matet

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  1. Pierre Matet
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Additional information

Part of this research was done while the author was visiting I.V.I.C. in Caracas in September 1989. The author would like to thank Carlos Di Prisco for his hospitality.

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Matet, P. Happy families and completely Ramsey sets. Arch Math Logic 32, 151–171 (1993). https://doi.org/10.1007/BF01375549

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  • DOI: https://doi.org/10.1007/BF01375549

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