Ideal version of Ramsey's theorem

Rafał Filipów; Nikodem Mrożek; Ireneusz Recław; Piotr Szuca

Czechoslovak Mathematical Journal (2011)

  • Volume: 61, Issue: 2, page 289-308
  • ISSN: 0011-4642

Abstract

top
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].

How to cite

top

Filipów, Rafał, et al. "Ideal version of Ramsey's theorem." Czechoslovak Mathematical Journal 61.2 (2011): 289-308. <http://eudml.org/doc/197028>.

@article{Filipów2011,
abstract = {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].},
author = {Filipów, Rafał, Mrożek, Nikodem, Recław, Ireneusz, Szuca, Piotr},
journal = {Czechoslovak Mathematical Journal},
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; ideal of subsets of natural numbers; Bolzano-Weierstrass theorem; Bolzano-Weierstrass property; ideal convergence; statistical density; statistical convergence; subsequence; monotone sequence; Ramsey's theorem},
language = {eng},
number = {2},
pages = {289-308},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Ideal version of Ramsey's theorem},
url = {http://eudml.org/doc/197028},
volume = {61},
year = {2011},
}

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
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 61
IS - 2
SP - 289
EP - 308
AB - 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].
LA - eng
KW - ideal of subsets of natural numbers; Bolzano-Weierstrass theorem; Bolzano-Weierstrass property; ideal convergence; statistical density; statistical convergence; subsequence; monotone sequence; Ramsey's theorem; ideal of subsets of natural numbers; Bolzano-Weierstrass theorem; Bolzano-Weierstrass property; ideal convergence; statistical density; statistical convergence; subsequence; monotone sequence; Ramsey's theorem
UR - http://eudml.org/doc/197028
ER -

References

top
  1. Alcántara, D. Meza, Ideals and filters on countable set, PhD thesis Universidad Nacional Autónoma de México (2009). (2009) 
  2. Baumgartner, J. E., Taylor, A. D., Wagon, S., Structural Properties of Ideals, Diss. Math. 197 (1982). (1982) Zbl0549.03036MR0687276
  3. Booth, D., 10.1016/0003-4843(70)90005-7, Ann. Math. Logic 2 (1970), 1-24. (1970) Zbl0231.02067MR0277371DOI10.1016/0003-4843(70)90005-7
  4. Burkill, H., Mirsky, L., 10.1016/0022-247X(73)90214-X, J. Math. Anal. Appl. 41 (1973), 391-410. (1973) Zbl0268.26007MR0335714DOI10.1016/0022-247X(73)90214-X
  5. Farah, I., 10.1112/S0025579300014054, Mathematika 45 (1998), 79-103. (1998) Zbl0903.03029MR1644345DOI10.1112/S0025579300014054
  6. Farah, I., Analytic Quotients: Theory of Liftings for Quotients over Analytic Ideals on the Integers, Mem. Am. Math. Soc 702 (2000). (2000) Zbl0966.03045MR1711328
  7. Filipów, R., Mrożek, N., Recław, I., Szuca, P., 10.2178/jsl/1185803621, J. Symb. Log. 72 (2007), 501-512. (2007) MR2320288DOI10.2178/jsl/1185803621
  8. Filipów, R., Szuca, P., 10.1016/j.jcta.2009.12.005, J. Comb. Theory, Ser. A 117 (2010), 943-956. (2010) Zbl1230.05036MR2652104DOI10.1016/j.jcta.2009.12.005
  9. Frankl, P., Graham, R. L., Rödl, V., 10.1016/0097-3165(90)90008-K, J. Comb. Theory, Ser. A 54 (1990), 95-111. (1990) MR1051781DOI10.1016/0097-3165(90)90008-K
  10. Kojman, M., 10.1090/S0002-9939-01-06116-0, Proc. Am. Math. Soc. 130 (2002), 631-635. (2002) Zbl0979.54036MR1866012DOI10.1090/S0002-9939-01-06116-0
  11. Mazur, K., 10.4064/fm-138-2-103-111, Fundam. Math. 138 (1991), 103-111. (1991) MR1124539DOI10.4064/fm-138-2-103-111
  12. Samet, N., Tsaban, B., 10.1016/j.topol.2009.04.014, Topology Appl. 156 (2009), 2659-2669. (2009) Zbl1231.05272MR2561218DOI10.1016/j.topol.2009.04.014
  13. Shelah, S., 10.1007/BFb0096536, Springer Berlin (1982). (1982) MR0675955DOI10.1007/BFb0096536
  14. Solecki, S., 10.1016/S0168-0072(98)00051-7, Ann. Pure Appl. Logic 99 (1999), 51-72. (1999) Zbl0932.03060MR1708146DOI10.1016/S0168-0072(98)00051-7
  15. Todorcevic, S., 10.1007/BFb0096295, Lecture Notes in Mathematics, Vol. 1652 Springer Berlin (1997). (1997) Zbl0953.54001MR1442262DOI10.1007/BFb0096295

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.