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ISSN 1088-6850(e) ISSN 0002-9947(p)

Block combinatorics

Authors: V. Farmaki and S. Negrepontis
Journal: Trans. Amer. Math. Soc. 358 (2006), 2759-2779
MSC (2000): Primary 03E05; Secondary 05D10, 46B20
Posted: January 27, 2006
MathSciNet review: 2204055
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Abstract: In this paper we extend the block combinatorics partition theorems of Hindman and Milliken-Taylor in the setting of the recursive system of the block Schreier families $(\mathcal{B}^\xi)$ , consisting of families defined for every countable ordinal $\xi$ . Results contain (a) a block partition Ramsey theorem for every countable ordinal $\xi$ (Hindman's Theorem corresponding to $\xi=1$ , and the Milliken-Taylor Theorem to $\xi$ a finite ordinal), (b) a countable ordinal form of the block Nash-Williams partition theorem, and (c) a countable ordinal block partition theorem for sets closed in the infinite block analogue of Ellentuck's topology.

References

[AA] D. Alspach and S. Argyros, Complexity of weakly null sequences, Dissertations Math. 321 (1992), 1-44. MR 1191024 (93j:46014)
[AO] D. Alspach and E. Odell, Averaging weakly null sequences, Lecture Notes in Math. 1332, Springer, Berlin, 1988, pp. 126-144. MR 0967092 (89j:46014)
[AGR] S. Argyros, G. Godefroy and H. Rosenthal, Descriptive set theory and Banach spaces, Handbook of the geometry of Banach spaces 2, North-Holland, Amsterdam (2003), 1007-1069. MR 1999190 (2004g:46002)
[B] J. Baumgartner, A short proof of Hindman's theorem, J. Combinatorial Theory, Ser. A, 17 (1974), 384-386. MR 0354394 (50:6873)
[E] E. Ellentuck, A new proof that analytic sets are Ramsey, J. Symb. Logic 39 (1974), 163-164. MR 0349393 (50:1887)
[F1] V. Farmaki, Classifications of Baire-1 functions and -spreading models, Trans. Amer. Math. Soc. 345 (2), (1994), 819-831. MR 1262339 (96c:46017)
[F2] V. Farmaki, Ramsey dichotomies with ordinal index, arXiv: math. LO/9804063 v1, (1998), electronic prepublication.
[F3] V. Farmaki, Ramsey and Nash-Williams combinatorics via Schreier families, arXiv: math. FA/0404014 v.1, (2004), electronic prepublication.
[F4] V. Farmaki, The uniform convergence ordinal index and the -behavior of a sequence of functions, Positivity 8 (1), (2004), 49-74. MR 2053575 (2005e:46029)
[G] W. T. Gowers, An infinite Ramsey theorem and some Banach space dichotomies, Annals of Mathematics 156 (2002), 797-833. MR 1954235 (2005a:46032)
[GRS] R. Graham, B. Rothschild and J. Spencer, Ramsey Theory, Wiley, New York, 1990. MR 1044995 (90m:05003)
[H] N. Hindman, Finite sums from sequences within cells of a partition of $\mathbb{N}$ , J. Combinatorial Theory, Ser. A 17 (1974), 1-11. MR 0349574 (50:2067)
[KS] J. Ketonen and R. Solovay, Rapidly growing Ramsey functions, Ann. of Math. 113 (1981), 267-314. MR 0607894 (84c:03100)
[K] K. Kuratowski, Topology, Volume I, Academic Press (1966). MR 0217751 (36:840)
[KM] K. Kuratowski and A. Mostowski, Set Theory, North-Holland, Amsterdam, 1968.
[L] A. Levy, Basic Set Theory, Springer-Verlag, 1979. MR 0533962 (80k:04001)
[M] K. Milliken, Ramsey's theorem with sums or unions, J. Combinatorial Theory, Ser. A 18 (1975), 276-290. MR 0373906 (51:10106)
[NW] C. St. J. A. Nash-Williams, On well quasiordering transfinite sequences, Proc. Camb. Phil. Soc. 61 (1965), 33-39. MR 0173640 (30:3850)
[O] E. Odell, On subspaces, asymptotic structure, and distortion of Banach spaces; connections with logic, Analysis and Logic, London Math. Soc. Lecture Note Ser. 262, Cambridge Univ. Press (2002), 189-267. MR 1967836 (2004b:46016)
[PH] J. Paris and L. Harrington, A mathematical incompleteness in Peano arithmetic, Handbook of Mathematical Logic, North-Holland, Amsterdam (1977), 1133-1142. MR 0457132 (56:15351)
[R] F. P. Ramsey, On a problem of formal logic, Proc. London Math. Soc. 30 (2), (1929), 264-286.
[S] J. Schreier, Ein Gegenbeispiel zur Theorie der schwachen Konvergenz, Studia Math. 2 (1930), 58-62.
[T1] A. Taylor, Some results in partition theory, Ph.D. dissertation, Dartmouth College, 1975.
[T2] A. Taylor, A canonical partition relation for finite subsets of $\omega$ , J. Combinatorial Theory, Ser. A 21 (1976), 137-146. MR 0424571 (54:12530)
[TJ] N. Tomczak-Jaegermann, Banach spaces of type have arbitrarily distortable subspaces, Geom. As. Funct. Anal. 6 (1996), 1074-1082. MR 1421875 (98g:46020)

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