Advertisement

Combinatorica

, Volume 39, Issue 1, pp 165–214 | Cite as

Reflection on the Coloring and Chromatic Numbers

  • Chris Lambie-HansonEmail author
  • Assaf Rinot
Article

Abstract

We prove that reflection of the coloring number of graphs is consistent with non-reflection of the chromatic number. Moreover, it is proved that incompactness for the chromatic number of graphs (with arbitrarily large gaps) is compatible with each of the following compactness principles: Rado’s conjecture, Fodor-type reflection, Δ-reflection, Stationary-sets reflection, Martin’s Maximum, and a generalized Chang’s conjecture. This is accomplished by showing that, under GCH-type assumptions, instances of incompactness for the chromatic number can be derived from square-like principles that are compatible with large amounts of compactness.

In addition, we prove that, in contrast to the chromatic number, the coloring number does not admit arbitrarily large incompactness gaps.

Mathematics Subject Classification (2010)

03E35 05C15 05C63 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    J. E. Baumgartner: Generic graph construction, J. Symbolic Logic 49 (1984), 234–240.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    A. M. Brodsky and A. Rinot: A microscopic approach to Souslin-tree constructions, Part I, Ann. Pure Appl. Logic, 168 (2017), 1949–2007.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    A. M. Brodsky and A. Rinot: A microscopic approach to Souslin-tree constructions, Part II, 2018, in preparation.zbMATHGoogle Scholar
  4. [4]
    A. M. Brodsky and A. Rinot: Distributive Aronszajn trees. Fund. Math., to appear 2019. http://www.assafrinot.com/paper/29 Google Scholar
  5. [5]
    J. Cummings and M. Magidor: Martin’s maximum and weak square, Proc. Amer. Math. Soc. 139 (2011), 3339–3348.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    J. Cummings: Iterated forcing and elementary embeddings, in: Handbook of set theory. Vols. 1, 2, 3, 775–883. Springer, Dordrecht, 2010.CrossRefGoogle Scholar
  7. [7]
    N. G. de Bruijn and P. Erdős: A colour problem for infinite graphs and a problem in the theory of relations, Nederl. Akad. Wetensch. Proc. Ser. A. 54 and Indagationes Math., 13 (1951), 369–373.MathSciNetzbMATHGoogle Scholar
  8. [8]
    P. Erdős and A. Hajnal: On chromatic number of infinite graphs, in: Theory of Graphs (Proc. Colloq., Tihany, 1966), 83–98. Academic Press, New York, 1968.Google Scholar
  9. [9]
    L. Fontanella and Y. Hayut: Square and delta Reflection, Ann. Pure Appl. Logic 167 (2016), 663–683.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    S. Fuchino, I. Juhász, L. Soukup, Z. Szentmiklóssy and T. Usuba: Fodor-type Reflection principle and Reflection of metrizability and meta-Lindelöfness, Topology Appl. 157 (2010), 1415–1429.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    M. Foreman and R. Laver: Some downwards transfer properties for N2, Adv. in Math. 67 (1988), 230–238.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    M. Foreman, M. Magidor and S. Shelah: Martin’s maximum, saturated ideals, and nonregular ultrafilters. I, Ann. of Math. (2) 127 (1988), 1–47.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    S. Fuchino, H. Sakai, L. Soukup and T. Usuba: More about the Fodor-type Reflection principle, preprint, 2012.Google Scholar
  14. [14]
    S. Fuchino, H. Sakai, V. T. Perez and T. Usuba: Rado’s Conjecture and the Fodor-type Reflection Principle, 2017, in preparation.Google Scholar
  15. [15]
    F. Galvin: Chromatic numbers of subgraphs, Period. Math. Hungar. 4 (1973), 117–119.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    Y. Hayut and C. Lambie-Hanson: Simultaneous stationary Reflection and square sequences, J. Math. Log., 17 (2017), 1750010.MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    A. Hajnal and A. Máté: Set mappings, partitions, and chromatic numbers, Studies in Logic and the Foundations of Mathematics 80 (1975), 347–379.CrossRefzbMATHGoogle Scholar
  18. [18]
    P. Komjáth: The colouring number, Proc. London Math. Soc. (3) 54 (1987), 1–14.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    P. Komjáth: Consistency results on infinite graphs, Israel J. Math. 61 (1988), 285–294.MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    R. Laver: Making the supercompactness of k indestructible under k-directed closed forcing, Israel J. Math. 29 (1978), 385–388.MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    C. Lambie-Hanson: Squares and covering matrices, Ann. Pure Appl. Logic 165 (2014), 673–694.MathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    T. Miyamoto: On the consistency strength of the FRP for the second uncountable cardinal, RIMS Kôkyûroku 1686 (2010), 80–92.Google Scholar
  23. [23]
    M. Magidor and S. Shelah: When does almost free imply free? (For groups, transversals, etc.), J. Amer. Math. Soc. 7 (1994), 769–830.MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    A. Rinot: Chain conditions of products, and weakly compact cardinals, Bull. Symb. Log. 20 (2014), 293–314.CrossRefzbMATHGoogle Scholar
  25. [25]
    A. Rinot: The Ostaszewski square, and homogeneous Souslin trees, Israel J. Math. 199 (2014), 975–1012.MathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    A. Rinot: Chromatic numbers of graphs - large gaps, Combinatorica 35 (2015), 215–233.MathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    A. Rinot: Hedetniemi’s conjecture for uncountable graphs, J. Eur. Math. Soc. (JEMS) 19 (2017), 285–298.MathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    A. Rinot: Higher Souslin trees and the GCH, revisited, Adv. Math. 311 (2017), 510–531.MathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    A. Rinot: Same graph, different universe, Arch. Math. Logic, 56 (2017), 783–796.MathSciNetCrossRefzbMATHGoogle Scholar
  30. [30]
    S. Shelah: Notes on partition calculus, in: Infinite and finite sets (Colloq., Keszthely, 1973; dedicated to P. Erdős on his 60th birthday), Vol. III, 1257–1276. Colloq. Math. Soc. János Bolyai, Vol. 10. North-Holland, Amsterdam, 1975.Google Scholar
  31. [31]
    S. Shelah: A compactness theorem for singular cardinals, free algebras, Whitehead problem and transversals, Israel J. Math. 21 (1975), 319–349.MathSciNetCrossRefzbMATHGoogle Scholar
  32. [32]
    S. Shelah: Incompactness for chromatic numbers of graphs, in: A tribute to Paul Erdős, 361–371, Cambridge Univ. Press, Cambridge, 1990.CrossRefGoogle Scholar
  33. [33]
    S. Shelah: Re ecting stationary sets and successors of singular cardinals, Arch. Math. Logic 31 (1991), 25–53.MathSciNetCrossRefzbMATHGoogle Scholar
  34. [34]
    S. Shelah: On incompactness for chromatic number of graphs, Acta Mathematica Hungarica 139 (2013), 363–371.MathSciNetCrossRefzbMATHGoogle Scholar
  35. [35]
    L. Soukup: On c+-chromatic graphs with small bounded subgraphs, Period. Math. Hungar. 21 (1990), 1–7.MathSciNetCrossRefzbMATHGoogle Scholar
  36. [36]
    S. Todorčević: On a conjecture of R. Rado, J. London Math. Soc. (2) 27 (1983), 1–8.MathSciNetzbMATHGoogle Scholar
  37. [37]
    S. Todorčević: A note on the proper forcing axiom, in: Axiomatic set theory (Boulder, Colo., 1983), volume 31 of Contemp. Math., 209–218. Amer. Math. Soc., Providence, RI, 1984.Google Scholar
  38. [38]
    S. Todorčević: Conjectures of Rado and Chang and cardinal arithmetic, in: Finite and infinite combinatorics in sets and logic (Banff, AB, 1991), volume 411 of NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 385–398. Kluwer Acad. Publ., Dordrecht, 1993.Google Scholar
  39. [39]
    S. Unger: Compactness for the chromatic number at Nω1+1, unpublished note, 2015.Google Scholar

Copyright information

© János Bolyai Mathematical Society and Springer-Verlag Berlin Heidelberg 2018

Authors and Affiliations

  1. 1.Department of MathematicsBar-Ilan UniversityRamat-GanIsrael

Personalised recommendations