Geodesic
Ideal version of Ramsey's theorem
Czechoslovak Mathematical Journal, Tome 61 (2011) no. 2, pp. 289-308

Voir la notice de l'article provenant de la source Czech Digital Mathematics Library

DOI MR   Zbl

We consider various forms of Ramsey's theorem, the monotone subsequence theorem and the Bolzano-Weierstrass theorem which are connected with ideals of subsets of natural numbers. We characterize ideals with properties considered. We show that, in a sense, Ramsey's theorem, the monotone subsequence theorem and the Bolzano-Weierstrass theorem characterize the same class of ideals. We use our results to show some versions of density Ramsey's theorem (these are similar to generalizations shown in [P. Frankl, R. L. Graham, and V. Rödl: Iterated combinatorial density theorems. J. Combin. Theory Ser. A 54 (1990), 95–111].
We consider various forms of Ramsey's theorem, the monotone subsequence theorem and the Bolzano-Weierstrass theorem which are connected with ideals of subsets of natural numbers. We characterize ideals with properties considered. We show that, in a sense, Ramsey's theorem, the monotone subsequence theorem and the Bolzano-Weierstrass theorem characterize the same class of ideals. We use our results to show some versions of density Ramsey's theorem (these are similar to generalizations shown in [P. Frankl, R. L. Graham, and V. Rödl: Iterated combinatorial density theorems. J. Combin. Theory Ser. A 54 (1990), 95–111].
DOI : 10.1007/s10587-011-0073-3
Classification : 05A17, 05D10, 11B05, 40A35, 54A20
Keywords: ideal of subsets of natural numbers; Bolzano-Weierstrass theorem; Bolzano-Weierstrass property; ideal convergence; statistical density; statistical convergence; subsequence; monotone sequence; Ramsey's theorem 
Filipów, Rafał; Mrożek, Nikodem; Recław, Ireneusz; Szuca, Piotr. Ideal version of Ramsey's theorem. Czechoslovak Mathematical Journal, Tome 61 (2011) no. 2, pp. 289-308. doi: 10.1007/s10587-011-0073-3
@article{10_1007_s10587_011_0073_3,
     author = {Filip\'ow, Rafa{\l} and Mro\.zek, Nikodem and Rec{\l}aw, Ireneusz and Szuca, Piotr},
     title = {Ideal version of {Ramsey's} theorem},
     journal = {Czechoslovak Mathematical Journal},
     pages = {289--308},
     year = {2011},
     volume = {61},
     number = {2},
     doi = {10.1007/s10587-011-0073-3},
     mrnumber = {2905404},
     zbl = {1249.05378},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1007/s10587-011-0073-3/}
}
TY  - JOUR
AU  - Filipów, Rafał
AU  - Mrożek, Nikodem
AU  - Recław, Ireneusz
AU  - Szuca, Piotr
TI  - Ideal version of Ramsey's theorem
JO  - Czechoslovak Mathematical Journal
PY  - 2011
SP  - 289
EP  - 308
VL  - 61
IS  - 2
UR  - http://geodesic.mathdoc.fr/articles/10.1007/s10587-011-0073-3/
DO  - 10.1007/s10587-011-0073-3
LA  - en
ID  - 10_1007_s10587_011_0073_3
ER  - 
%0 Journal Article
%A Filipów, Rafał
%A Mrożek, Nikodem
%A Recław, Ireneusz
%A Szuca, Piotr
%T Ideal version of Ramsey's theorem
%J Czechoslovak Mathematical Journal
%D 2011
%P 289-308
%V 61
%N 2
%U http://geodesic.mathdoc.fr/articles/10.1007/s10587-011-0073-3/
%R 10.1007/s10587-011-0073-3
%G en
%F 10_1007_s10587_011_0073_3

[1] Alcántara, D. Meza: Ideals and filters on countable set. PhD thesis Universidad Nacional Autónoma de México (2009).

[2] Baumgartner, J. E., Taylor, A. D., Wagon, S.: Structural Properties of Ideals. Diss. Math. 197 (1982). | MR | Zbl

[3] Booth, D.: Ultrafilters on a countable set. Ann. Math. Logic 2 (1970), 1-24. | DOI | MR | Zbl

[4] Burkill, H., Mirsky, L.: Monotonicity. J. Math. Anal. Appl. 41 (1973), 391-410. | DOI | MR | Zbl

[5] Farah, I.: Semiselective coideals. Mathematika 45 (1998), 79-103. | DOI | MR | Zbl

[6] Farah, I.: Analytic Quotients: Theory of Liftings for Quotients over Analytic Ideals on the Integers. Mem. Am. Math. Soc 702 (2000). | MR | Zbl

[7] Filipów, R., Mrożek, N., Recław, I., Szuca, P.: Ideal convergence of bounded sequences. J. Symb. Log. 72 (2007), 501-512. | DOI | MR

[8] Filipów, R., Szuca, P.: Density versions of Schur's theorem for ideals generated by submeasures. J. Comb. Theory, Ser. A 117 (2010), 943-956. | DOI | MR | Zbl

[9] Frankl, P., Graham, R. L., Rödl, V.: Iterated combinatorial density theorems. J. Comb. Theory, Ser. A 54 (1990), 95-111. | DOI | MR

[10] Kojman, M.: Van der Waerden spaces. Proc. Am. Math. Soc. 130 (2002), 631-635. | DOI | MR | Zbl

[11] Mazur, K.: $F_\sigma$-ideals and $\omega_1\omega_1^*$-gaps in the Boolean algebras {$P(\omega)/I$}. Fundam. Math. 138 (1991), 103-111. | DOI | MR

[12] Samet, N., Tsaban, B.: Superfilters, Ramsey theory, and van der Waerden's theorem. Topology Appl. 156 (2009), 2659-2669. | DOI | MR | Zbl

[13] Shelah, S.: Proper Forcing. Lecture Notes in Mathematics, Vol. 940. Springer Berlin (1982). | DOI | MR

[14] Solecki, S.: Analytic ideals and their applications. Ann. Pure Appl. Logic 99 (1999), 51-72. | DOI | MR | Zbl

[15] Todorcevic, S.: Topics in Topology. Lecture Notes in Mathematics, Vol. 1652 Springer Berlin (1997). | DOI | MR | Zbl

Cité par Sources :