A discretized approach to W. T. Gowers' game

• Autorzy: Kanellopoulos V. , Tyros K.
• Języki publikacji: EN

Abstrakty

EN

We give an alternative proof of W. T. Gowers' theorem on block bases by reducing it to a discrete analogue on specific countable nets. We also give a Ramsey type result on k-tuples of block sequences in a normed linear space with a Schauder basis.

Słowa kluczowe

EN

Ramsey theory; games in Banach spaces

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2010
• Tom: Vol. 58, no 1
• Strony: 1--16
• Opis fizyczny: Bibliogr. 24 poz.

Twórcy

autor

Kanellopoulos V.

autor

Tyros K.

• Department of Mathematics, Faculty of Applied Sciences, National Technical University of Athens, Zografou Campus 157 80, Athens, Greece,

Bibliografia

• [1] G. Androulakis, S. J. Dilworth, and N. J. Kalton, A new approach to the Ramsey-type games and the Gowers dichotomy in F-spaces, Combinatorica, to appear.
• [2] S. A. Argyros and S. Todorčevič, Ramsey Methods in Analysis, Adv. Courses Math., CRM Barcelona, Birkhäuser, Basel, 2005.
• [3] J. Bagaria and J. Lopez-Abad, Weakly Ramsey sets in Banach spaces, Adv. Math. 160 (2001), 133-174.
• [4] J. Bagaria and J. Lopez-Abad, Determinacy and weakly Ramsey sets in Banach spaces, Trans. Amer. Math. Soc. 354 (2002), 1327-1349.
• [5] E. Ellentuck, A new proof that analytic sets are Ramsey, J. Symbolic Logic 39 (1974), 163-165.
• [6] V. Ferenczi and C. Rosendal, Banach spaces without minimal subspaces, J. Funct. Anal. 257 (2009), 149-193.
• [7] T. Figiel, R. Frankiewicz, R. Komorowski and C. Ryll-Nardzewski, On hereditarily indecomposable Banach spaces, Ann. Pure Appl. Logic 126 (2004), 293-299.
• [8] T. Figiel, R. Frankiewicz, R. Komorowski and C. Ryll-Nardzewski, Selecting basic sequences in φ-stable Banach spaces, Studia Math. 159 (2003), 499-515.
• [9] F. Galvin and K. Prikry, Borel sets and Ramsey’s theorem, J. Symbolic Logic 38 (1973), 193-198.
• [10] W. T. Gowers, A new dichotomy for Banach spaces, Geom. Funct. Anal. 6 (1996), 1083-1093.
• [11] W. T. Gowers, An infinite Ramsey theorem and some Banach-space dichotomies, Ann. of Math. 156 (2002), 797-833.
• [12] W. T. Gowers, Ramsey methods in Banach spaces, in: Handbook of the Geometry of Banach Spaces, Vol. 2, Elsevier, 2003, 1072-1097.
• [13] A. S. Kechris, Classical Descriptive Set Theory, Springer, 1995.
• [14] J. Lopez-Abad, Coding into Ramsey sets, Math. Ann. 332 (2005), 775-794.
• [15] B. Maurey, A note on Cowers’ dichotomy theorem, in: Convex Geometric Analysis, Math. Sci. Res. Inst. Publ. 34, Cambridge Univ. Press, Cambridge, 1999, 149-157.
• [16] K. Milliken, Ramsey’s theorem with sums and unions, J. Combin. Theory Ser. A 18 (1975), 276-290.
• [17] C. St. J. A. Nash-Williams, On well-quasi-ordering transfinite sequences, Proc. Cambridge Philos. Soc. 61 (1965), 33-39.
• [18] A. M. Pelczar, Some version of Gowers’ dichotomy for Banach spaces, Univ. lagel. Acta Math. 41 (2003), 235-243.
• [19] A. M. Pelczar, Subsymmetric sequences and minimal spaces, Proc. Amer. Math. Soc. 131 (2003), 765-771.
• [20] P. Pudlák and V. Rödl, Partition theorems for systems of finite subsets of integers, Discrete Math. 39 (1982), 67-73.
• [21] C. Rosendal, An exact Ramsey principle for block sequences, Collect. Math. 61 (2010), 25-36.
• [22] C. Rosendal, Infinite asymptotic games, Ann. Inst. Fourier (Grenoble) 59 (2009), 1323-1348.
• [23] J. Silver, Every analytic set is Ramsey, J. Symbolic Logic 35 (1970), 60-64.
• [24] A. Tcaciuc, On the existence of asymptotic-lp structures in Banach spaces, Canad. Math. Bull. 50 (2007), 619-631.