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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Forcing positive partition relations
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by Stevo Todorčević PDF
Trans. Amer. Math. Soc. 280 (1983), 703-720 Request permission


We show how to force two strong positive partition relations on ${\omega _1}$ and use them in considering several well-known open problems.
  • Uri Avraham and Saharon Shelah, Martin’s axiom does not imply that every two $\aleph _{1}$-dense sets of reals are isomorphic, Israel J. Math. 38 (1981), no. 1-2, 161–176. MR 599485, DOI 10.1007/BF02761858
  • —, Isomorphism types of Aronszajn trees (to appear). U. Avraham, M. Rubin and S. Shelah, On the consistency of some partition theorems for continuous colorings, and the structure of ${\aleph _1}$-dense real order types (to appear). U. Avraham and S. Todorčević. Martin’s axiom and first countable $S$ and $L$ spaces, Handbook of the Set-Theoretic Topology (to appear).
  • James E. Baumgartner, Iterated forcing, Surveys in set theory, London Math. Soc. Lecture Note Ser., vol. 87, Cambridge Univ. Press, Cambridge, 1983, pp. 1–59. MR 823775, DOI 10.1017/CBO9780511758867.002
  • —, Application of the proper forcing axiom, Handbook of the Set-Theoretic Topology (to appear). K. J. Devlin, The Yorkshireman’s guide to proper forcing (to appear).
  • Keith J. Devlin and Håvard Johnsbråten, The Souslin problem, Lecture Notes in Mathematics, Vol. 405, Springer-Verlag, Berlin-New York, 1974. MR 0384542, DOI 10.1007/BFb0065979
  • Ben Dushnik and E. W. Miller, Partially ordered sets, Amer. J. Math. 63 (1941), 600–610. MR 4862, DOI 10.2307/2371374
  • B. Efimov, On the power of Hausdorff spaces, Dokl. Akad. Nauk SSSR 164 (1965), 967–970 (Russian). MR 0190891
  • P. Erdös and R. Rado, A partition calculus in set theory, Bull. Amer. Math. Soc. 62 (1956), 427–489. MR 81864, DOI 10.1090/S0002-9904-1956-10036-0
  • P. Erdős and A. Hajnal, Unsolved problems in set theory, Axiomatic Set Theory (Proc. Sympos. Pure Math., Vol. XIII, Part I, Univ. California, Los Angeles, Calif., 1967) Amer. Math. Soc., Providence, R.I., 1971, pp. 17–48. MR 0280381
  • P. Erdős and A. Hajnal, Unsolved and solved problems in set theory, Proceedings of the Tarski Symposium (Proc. Sympos. Pure Math., Vol. XXV, Univ. California, Berkeley, Calif., 1971) Amer. Math. Soc., Providence, R.I., 1974, pp. 269–287. MR 0357122
  • V. Fedorčuk, On the cardinality of hereditarily separable compuct Hausdorff spaces, Soviet Math. Dokl. 16 (1975), 651-655. F. Galvin, On Gruenhage’s generalization of first countable spaces. II, Notices Amer. Math. Soc. 24 (1977), A-257.
  • J. de Groot, Discrete subspaces of Hausdorff spaces, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 13 (1965), 537–544. MR 210061
  • A. Hajnal, Some results and problems on set theory, Acta Math. Acad. Sci. Hungar. 11 (1960), 277–298 (English, with Russian summary). MR 150044, DOI 10.1007/BF02020945
  • A. Hajnal and I. Juhász, Discrete subspaces of topological spaces, Nederl. Akad. Wetensch. Proc. Ser. A 70=Indag. Math. 29 (1967), 343–356. MR 0229195
  • A. Hajnal and I. Juhász, A consistency result concerning hereditarily$\alpha$-separable spaces, Nederl. Akad. Wetensch. Proc. Ser. A 76=Indag. Math. 35 (1973), 301–307. MR 0362187, DOI 10.1016/1385-7258(73)90025-5
  • John R. Isbell, Remarks on spaces of large cardinal number, Czechoslovak Math. J. 14(89) (1964), 383–385 (English, with Russian summary). MR 177383, DOI 10.21136/CMJ.1964.100627
  • Thomas Jech, Set theory, Pure and Applied Mathematics, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 506523
  • I. Juhász, A survey of $S$- and $L$-spaces, Topology, Vol. II (Proc. Fourth Colloq., Budapest, 1978) Colloq. Math. Soc. János Bolyai, vol. 23, North-Holland, Amsterdam-New York, 1980, pp. 675–688. MR 588816
  • Kenneth Kunen, Set theory, Studies in Logic and the Foundations of Mathematics, vol. 102, North-Holland Publishing Co., Amsterdam-New York, 1980. An introduction to independence proofs. MR 597342
  • R. Laver, Partition relations for uncountable cardinals $\leq 2^{\aleph _{0}}$, Infinite and finite sets (Colloq., Keszthely, 1973; dedicated to P. Erdős on his 60th birthday), Vol. II, Colloq. Math. Soc. János Bolyai, Vol. 10, North-Holland, Amsterdam, 1975, pp. 1029–1042. MR 0371652
  • Richard Laver, On the consistency of Borel’s conjecture, Acta Math. 137 (1976), no. 3-4, 151–169. MR 422027, DOI 10.1007/BF02392416
  • William Mitchell, Aronszajn trees and the independence of the transfer property, Ann. Math. Logic 5 (1972/73), 21–46. MR 313057, DOI 10.1016/0003-4843(72)90017-4
  • F. P. Ramsey, On the problem of formal logic. Proc. London Math. Soc. (2) 30 (1930), 264-286.
  • Judy Roitman, Basic $S$ and $L$, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 295–326. MR 776626
  • Mary Ellen Rudin, A normal hereditarily separable non-Lindelöf space, Illinois J. Math. 16 (1972), 621–626. MR 309062
  • Mary Ellen Rudin, $S$ and $L$ spaces, Surveys in general topology, Academic Press, New York-London-Toronto, Ont., 1980, pp. 431–444. MR 564109
  • Saharon Shelah, Proper forcing, Lecture Notes in Mathematics, vol. 940, Springer-Verlag, Berlin-New York, 1982. MR 675955, DOI 10.1007/978-3-662-21543-2
  • W. Sierpiński, Sur un problème de la théorie des relutions, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (2) 2 (1933), 285-287.
  • R. M. Solovay and S. Tennenbaum, Iterated Cohen extensions and Souslin’s problem, Ann. of Math. (2) 94 (1971), 201–245. MR 294139, DOI 10.2307/1970860
  • S. Todorčević, On the $S$-space problem, Abstracts Amer. Math. Soc. 2 (1981), no. 4, A-394. —, ${\omega _1} \to {({\omega _1},\omega + 2)^2}$ is consistent, Abstracts Amer. Math. Soc. 2 (1981), no. 5, A-462. —, On the cardinality of Hausdorff spaces, Abstracts Amer. Math. Soc. 2 (1981), no. 6, A-529. N. H. Williams, Combinatorial set theory, North-Holland, Amsterdam, 1977.
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Additional Information
  • © Copyright 1983 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 280 (1983), 703-720
  • MSC: Primary 03E35; Secondary 03C62, 03E05, 54A35
  • DOI:
  • MathSciNet review: 716846