×

Homogeneity for open partitions of pairs of reals. (English) Zbl 0795.03065

It is shown that for every analytic set \(A\subset\mathbb{R}\) and every partition of the two-element subsets of \(A\), \([A]^ 2= K_ 0\cup K_ 1\), where \(K_ 0\) is open, either there exists a perfect set \(P\) (homeomorphic copy of the Cantor set) which is 0-homogeneous \(([P^ 2]\subseteq K_ 0)\), or \(A\) is the countable union of 1-homogeneous sets. This is a definable version of the Open Coloring Axiom introduced by Todorcevic. It is also shown that the above statement for coanalytic sets in place of analytic sets is equivalent to every uncountable coanalytic set of reals contains a perfect subset. It is proved that every set of reals has this property in Solovay’s model where every set of reals is measurable. It is also shown that PD (every projective set of reals is determined) implies this partition result for every projective set of reals.

MSC:

03E15 Descriptive set theory
03E05 Other combinatorial set theory
03E60 Determinacy principles
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Uri Abraham, Matatyahu Rubin, and Saharon Shelah, On the consistency of some partition theorems for continuous colorings, and the structure of ℵ\(_{1}\)-dense real order types, Ann. Pure Appl. Logic 29 (1985), no. 2, 123 – 206. · Zbl 0585.03019 · doi:10.1016/0168-0072(84)90024-1
[2] James E. Baumgartner, Applications of the proper forcing axiom, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 913 – 959. · Zbl 0556.03040
[3] Andreas Blass, A partition theorem for perfect sets, Proc. Amer. Math. Soc. 82 (1981), no. 2, 271 – 277. · Zbl 0472.03038
[4] Morton Davis, Infinite games of perfect information, Advances in game theory, Princeton Univ. Press, Princeton, N.J., 1964, pp. 85 – 101. · Zbl 0133.13104
[5] A. Dodd and R. Jensen, The core model, Ann. Math. Logic 20 (1981), no. 1, 43 – 75. · Zbl 0457.03051 · doi:10.1016/0003-4843(81)90011-5
[6] P. Erdös and R. Rado, A partition calculus in set theory, Bull. Amer. Math. Soc. 62 (1956), 427 – 489. · Zbl 0071.05105
[7] F. M. Filipczak, Sur les fonctions continues relativement monotones, Fund. Math. 58 (1966), 75 – 87 (French). · Zbl 0185.12204
[8] F. Galvin, Partition theorems for the real line, Notices Amer. Math. Soc. 15 (1968), 660.
[9] Leo Harrington, David Marker, and Saharon Shelah, Borel orderings, Trans. Amer. Math. Soc. 310 (1988), no. 1, 293 – 302. · Zbl 0707.03042
[10] Thomas Jech, Set theory, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. Pure and Applied Mathematics. · Zbl 0419.03028
[11] D. Donder, R. B. Jensen, and B. J. Koppelberg, Some applications of the core model, Set theory and model theory (Bonn, 1979) Lecture Notes in Math., vol. 872, Springer, Berlin-New York, 1981, pp. 55 – 97.
[12] Richard Mansfield, Perfect subsets of definable sets of real numbers, Pacific J. Math. 35 (1970), 451 – 457. · Zbl 0251.02060
[13] Donald A. Martin and John R. Steel, Projective determinacy, Proc. Nat. Acad. Sci. U.S.A. 85 (1988), no. 18, 6582 – 6586. · Zbl 0656.03036 · doi:10.1073/pnas.85.18.6582
[14] Donald A. Martin and John R. Steel, A proof of projective determinacy, J. Amer. Math. Soc. 2 (1989), no. 1, 71 – 125. · Zbl 0668.03021
[15] Jan Mycielski, Independent sets in topological algebras, Fund. Math. 55 (1964), 139 – 147. · Zbl 0124.01301
[16] F. Ramsey, On a problem of formal logic, Proc. London Math. Soc. 30 (1930), 264-286. · JFM 55.0032.04
[17] W. Sierpiński, Sur un problème de la théorie des relations, Ann. Scuola Norm. Sup. Pisa 2 (1937), 285-287. · JFM 59.0092.01
[18] J. R. Shoenfield, The problem of predicativity, Essays on the foundations of mathematics, Magnes Press, Hebrew Univ., Jerusalem, 1961, pp. 132 – 139.
[19] Robert M. Solovay, On the cardinality of \sum \(_{2}\)\textonesuperior sets of reals, Foundations of Mathematics (Symposium Commemorating Kurt Gödel, Columbus, Ohio, 1966) Springer, New York, 1969, pp. 58 – 73.
[20] Stevo Todorčević, Partition problems in topology, Contemporary Mathematics, vol. 84, American Mathematical Society, Providence, RI, 1989. · Zbl 0659.54001
[21] -, Two examples of Borel partially ordered sets with countable chain conditions, preprint, 1990.
[22] -, Conference talk, MSRI, October 1989.
[23] B. Velickovic, Conference talk, MSRI, October 1989.
[24] W. Hugh Woodin, Supercompact cardinals, sets of reals, and weakly homogeneous trees, Proc. Nat. Acad. Sci. U.S.A. 85 (1988), no. 18, 6587 – 6591. · Zbl 0656.03037 · doi:10.1073/pnas.85.18.6587
[25] P. Erdős, K. Kunen, and R. Daniel Mauldin, Some additive properties of sets of real numbers, Fund. Math. 113 (1981), no. 3, 187 – 199. · Zbl 0482.28001
[26] David Gale and F. M. Stewart, Infinite games with perfect information, Contributions to the theory of games, vol. 2, Annals of Mathematics Studies, no. 28, Princeton University Press, Princeton, N. J., 1953, pp. 245 – 266. · Zbl 0050.14305
[27] Robert M. Solovay, A model of set-theory in which every set of reals is Lebesgue measurable, Ann. of Math. (2) 92 (1970), 1 – 56. · Zbl 0207.00905 · doi:10.2307/1970696
[28] Arnold W. Miller, Special subsets of the real line, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 201 – 233. · Zbl 0588.54035
[29] R. M. Solovay and S. Tennenbaum, Iterated Cohen extensions and Souslin’s problem, Ann. of Math. (2) 94 (1971), 201 – 245. · Zbl 0244.02023 · doi:10.2307/1970860
[30] Donald A. Martin, Borel determinacy, Ann. of Math. (2) 102 (1975), no. 2, 363 – 371. · Zbl 0336.02049 · doi:10.2307/1971035
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.