More results in polychromatic Ramsey theory

Uri Abraham; James Cummings

Open Mathematics (2012)

  • Volume: 10, Issue: 3, page 1004-1016
  • ISSN: 2391-5455

Abstract

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We study polychromatic Ramsey theory with a focus on colourings of [ω 2]2. We show that in the absence of GCH there is a wide range of possibilities. In particular each of the following is consistent relative to the consistency of ZFC: (1) 2ω = ω 2 and ω 2 → p o l y ( α ) ℵ 0 - b d d 2 for every α <ω 2; (2) 2ω = ω 2 and ω 2 ↛ p o l y ( ω 1 ) 2 - b d d 2 .

How to cite

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Uri Abraham, and James Cummings. "More results in polychromatic Ramsey theory." Open Mathematics 10.3 (2012): 1004-1016. <http://eudml.org/doc/269526>.

@article{UriAbraham2012,
abstract = {We study polychromatic Ramsey theory with a focus on colourings of [ω 2]2. We show that in the absence of GCH there is a wide range of possibilities. In particular each of the following is consistent relative to the consistency of ZFC: (1) 2ω = ω 2 and $\omega _2 \rightarrow ^\{poly\} (\alpha )_\{\aleph _0 - bdd\}^2 $ for every α <ω 2; (2) 2ω = ω 2 and $\omega _2 \nrightarrow ^\{poly\} (\omega _1 )_\{2 - bdd\}^2 $ .},
author = {Uri Abraham, James Cummings},
journal = {Open Mathematics},
keywords = {Polychromatic Ramsey theory; Proper forcing; Models as side conditions; polychromatic Ramsey theory; proper forcing; models as side conditions},
language = {eng},
number = {3},
pages = {1004-1016},
title = {More results in polychromatic Ramsey theory},
url = {http://eudml.org/doc/269526},
volume = {10},
year = {2012},
}

TY - JOUR
AU - Uri Abraham
AU - James Cummings
TI - More results in polychromatic Ramsey theory
JO - Open Mathematics
PY - 2012
VL - 10
IS - 3
SP - 1004
EP - 1016
AB - We study polychromatic Ramsey theory with a focus on colourings of [ω 2]2. We show that in the absence of GCH there is a wide range of possibilities. In particular each of the following is consistent relative to the consistency of ZFC: (1) 2ω = ω 2 and $\omega _2 \rightarrow ^{poly} (\alpha )_{\aleph _0 - bdd}^2 $ for every α <ω 2; (2) 2ω = ω 2 and $\omega _2 \nrightarrow ^{poly} (\omega _1 )_{2 - bdd}^2 $ .
LA - eng
KW - Polychromatic Ramsey theory; Proper forcing; Models as side conditions; polychromatic Ramsey theory; proper forcing; models as side conditions
UR - http://eudml.org/doc/269526
ER -

References

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  1. [1] Abraham U., Cummings J., Smyth C., Some results in polychromatic Ramsey theory, J. Symbolic Logic, 2007, 72(3), 865–896 http://dx.doi.org/10.2178/jsl/1191333845 Zbl1125.03034
  2. [2] Baumgartner J., Hajnal A., A proof (involving Martin’s axiom) of a partition relation, Fund. Math., 1973, 78(3), 193–203 Zbl0257.02054
  3. [3] Galvin F., Advanced problems: 6034, Amer. Math. Monthly, 1975, 82(5), 529 
  4. [4] Galvin F., letters to S. Todorčevic, cited in: Todorčevic S., Forcing positive partition relations, Trans. Amer. Math. Soc., 1983, 280(2), 703–720 
  5. [5] Koszmider P., On strong chains of uncountable functions, Israel J. Math., 2000, 118, 289–315 http://dx.doi.org/10.1007/BF02803525 Zbl0961.03039
  6. [6] Shelah S., Zapletal J., Embeddings of Cohen algebras, Adv. Math., 1997, 126(2), 93–118 http://dx.doi.org/10.1006/aima.1996.1597 
  7. [7] Todorčević S., Forcing positive partition relations, Trans. Amer. Math. Soc., 1983, 280(2), 703–720 Zbl0532.03023
  8. [8] Todorčević S., A note on the proper forcing axiom, In: Axiomatic Set Theory, Boulder, June 19–25, 1983, Contemp. Math. 31, American Mathematical Society, Providence, 1984 
  9. [9] Todorčević S., Directed sets and cofinal types, Trans. Amer. Math. Soc., 1985, 290(2), 711–723 Zbl0592.03037

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