A dichotomy theorem for subsets of the power set of the natural numbers

Gasparis, I.

doi:10.1090/S0002-9939-00-05594-5

A dichotomy theorem for subsets of the power set of the natural numbers
HTML articles powered by AMS MathViewer

by I. Gasparis PDF

Proc. Amer. Math. Soc. 129 (2001), 759-764 Request permission

Abstract:

The following dichotomy is established for any pair $\mathcal {F}$, $\mathcal {G}$ of hereditary families of finite subsets of $\mathbb {N}$: Given $N$, an infinite subset of $\mathbb {N}$, there exists $M$ an infinite subset of $N$ so that either $\mathcal {G} \cap [M]^{< \infty } \subset \mathcal {F}$, or $\mathcal {F} \cap [M]^{< \infty } \subset \mathcal {G}$, where $[M]^{< \infty }$ denotes the set of all finite subsets of $M$.

References

Dale E. Alspach and Spiros Argyros, Complexity of weakly null sequences, Dissertationes Math. (Rozprawy Mat.) 321 (1992), 44. MR 1191024
G. Androulakis and E. Odell, Distorting mixed Tsirelson spaces, Israel J. Math. 109 (1999), 125–149.
S. A. Argyros and I. Deliyanni, Examples of asymptotic $l_1$ Banach spaces, Trans. Amer. Math. Soc. 349 (1997), no. 3, 973–995. MR 1390965, DOI 10.1090/S0002-9947-97-01774-1
S.A. Argyros, S. Mercourakis and A. Tsarpalias, Convex unconditionality and summability of weakly null sequences, Israel J. Math. 107 (1998), 157–193.
S.A. Argyros and V. Felouzis, Interpolating hereditarily indecomposable Banach spaces, (preprint).
S.A. Argyros and I. Gasparis, Unconditional structures of weakly null sequences, (preprint).
Erik Ellentuck, A new proof that analytic sets are Ramsey, J. Symbolic Logic 39 (1974), 163–165. MR 349393, DOI 10.2307/2272356
Fred Galvin and Karel Prikry, Borel sets and Ramsey’s theorem, J. Symbolic Logic 38 (1973), 193–198. MR 337630, DOI 10.2307/2272055
V. Farmaki, Ramsey dichotomies with ordinal index, (preprint)
R. Judd, A dichotomy on Schreier sets, Studia Math. 132 (1999), 245–256.
P. Kyriakouli, On hereditary families of finite subsets of positive integers, (preprint).
C. St. J. A. Nash-Williams, On well-quasi-ordering transfinite sequences, Proc. Cambridge Philos. Soc. 61 (1965), 33–39. MR 173640, DOI 10.1017/s0305004100038603
S. Mazurkiewicz and W. Sierpinski, Contribution a la topologie des ensembles denomrables, Fund. Math. 3 (1920), 17–27.
E. Odell, Applications of Ramsey theorems to Banach space theory, Notes in Banach spaces, Univ. Texas Press, Austin, Tex., 1980, pp. 379–404. MR 606226
E. Odell and Th. Schlumprecht, A problem on spreading models, J. Funct. Anal. 153 (1998), no. 2, 249–261. MR 1614578, DOI 10.1006/jfan.1997.3191
E. Odell, N. Tomczak-Jaegermann, R. Wagner, Proximity to $\ell _1$ and Distortion in asymptotic $\ell _1$ spaces, J. Funct. Anal. 150 (1997), no. 1, 101–145.
J. Schreier, Ein Gegenbeispiel zur theorie der schwachen konvergenz, Studia Math. 2 (1930), 58–62.
Jack Silver, Every analytic set is Ramsey, J. Symbolic Logic 35 (1970), 60–64. MR 332480, DOI 10.2307/2271156
A. Tsarpalias, A note on the Ramsey property, Proc. Amer. Math. Soc. 127 (1999), no. 2, 583–587. MR 1458267, DOI 10.1090/S0002-9939-99-04518-9