Strong sequences and partition relations

Joanna Jureczko

Abstract


The first result in partition relations topic belongs to Ramsey (1930). Since that this topic has been still explored. Probably the most famous partition theorem is Erdős-Rado theorem (1956). On the other hand in 60's of the last century Efimov introduced strong sequences method, which was used for proving some famous theorems in dyadic spaces. The aim of this paper is to generalize theorem on strong sequences and to show that it is equivalent to generalized version of well-known Erdős-Rado theorem. It will be also shown that this equivalence holds for singulars. Some applications and conclusions will be presented too.

Keywords


Strong sequences; Erdős-Rado theorem; partition relations; inaccessible cardinals; singular cardinals

Mathematics Subject Classification (2010)


03E02; 03E05; 03E20; 54A25

References


Baumgartner, James E., and Alan D. Taylor, and Stanley Wagon. Structural properties of ideals. Vol 197 of Dissertationes Math. (Rozprawy Mat.). Warszawa: PWN–Polish Scientific Publishers, 1982.

Comfort, W. Wistar, and Stylianos A. Negrepontis. Chain conditions in topology. Vol 79 of Cambridge Tracts in Mathematics. Cambridge-New York: Cambridge University Press, 1982.

Efimov, B. A. "Dyadic bicompacta." (Russian) Trudy Moskov. Mat. Obšc. 14 (1965): 211–247.

Engelking, Ryszard. General topology. Vol. 60 of Monografie Matematyczne. Warszawa: PWN–Polish Scientific Publishers, 1977.

Erdös, Paul, and Richard Rado. "A partition calculus in set theory." Bull. Amer. Math. Soc. 62 (1956): 427–489.

Hajnal, András. "Proof of a conjecture of S. Ruziewicz." Fund. Math. 50 (1961/1962): 123–128.

Jech, Thomas. Set theory. New York-London: Academic Press, 1978.

Jureczko, Joanna. "On inequalities among some cardinal invariants." Math. Aeterna 6, no. 1 (2016): 87–98.

Jureczko, Joanna. "Strong sequences and independent sets." Math. Aeterna 6, no. 2 (2016): 141–152.

Jureczko, Joanna. "kappa-strong sequences and the existence of generalized independent families." Open Mathematics (2017) (to appear).

Szpilrajn, Edward. "Remarque sur les produits cartésiens d’espaces topologiques." C. R. (Doklady) Acad. Sci. URSS (N. S.) 31 (1941): 525–527.

Turzanski, Marian. "Strong sequences and the weight of regular spaces." Comment. Math. Univ. Carolin. 33, no. 3 (1992): 557–561.

Turzanski, Marian. "On the selector of twin functions." Comment. Math. Univ. Carolin. 39, no. 2 (1998): 303–307.


Full Text: PDF

Download statistics: 1306

Licencja Creative Commons
This article by Joanna Jureczko is governed by the Creative Commons Attribution-ShareAlike 4.0 International(CC BY-SA 4.0) licence.


e-ISSN: 2300-133X, ISSN: 2081-545X

Since 2017 Open Access in De Gruyter and CrossCheck access cofinanced by The Ministry of Science and Higher Education - Republic of Poland - DUN 775/P-DUN/2017 see more

The Journal is indexed in:
and others see Abstracting and Indexing list

AUPC SM is on the List of the Ministry’s scored journals with 20 points for 2019

Deklaracja dostępności cyfrowej