Reflection on the Coloring and Chromatic Numbers

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.

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References

  1. [1]

    J. E. Baumgartner: Generic graph construction, J. Symbolic Logic 49 (1984), 234–240.

    MathSciNet  Article  MATH  Google 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.

    MathSciNet  Article  MATH  Google Scholar 

  3. [3]

    A. M. Brodsky and A. Rinot: A microscopic approach to Souslin-tree constructions, Part II, 2018, in preparation.

    Google 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.

    MathSciNet  Article  MATH  Google Scholar 

  6. [6]

    J. Cummings: Iterated forcing and elementary embeddings, in: Handbook of set theory. Vols. 1, 2, 3, 775–883. Springer, Dordrecht, 2010.

    Google 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.

    MathSciNet  MATH  Google 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.

    MathSciNet  Article  MATH  Google 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.

    MathSciNet  Article  MATH  Google Scholar 

  11. [11]

    M. Foreman and R. Laver: Some downwards transfer properties for N2, Adv. in Math. 67 (1988), 230–238.

    MathSciNet  Article  MATH  Google 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.

    MathSciNet  Article  MATH  Google 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.

    MathSciNet  Article  MATH  Google Scholar 

  16. [16]

    Y. Hayut and C. Lambie-Hanson: Simultaneous stationary Reflection and square sequences, J. Math. Log., 17 (2017), 1750010.

    MathSciNet  Article  MATH  Google 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.

    Article  MATH  Google Scholar 

  18. [18]

    P. Komjáth: The colouring number, Proc. London Math. Soc. (3) 54 (1987), 1–14.

    MathSciNet  Article  MATH  Google Scholar 

  19. [19]

    P. Komjáth: Consistency results on infinite graphs, Israel J. Math. 61 (1988), 285–294.

    MathSciNet  Article  MATH  Google Scholar 

  20. [20]

    R. Laver: Making the supercompactness of k indestructible under k-directed closed forcing, Israel J. Math. 29 (1978), 385–388.

    MathSciNet  Article  MATH  Google Scholar 

  21. [21]

    C. Lambie-Hanson: Squares and covering matrices, Ann. Pure Appl. Logic 165 (2014), 673–694.

    MathSciNet  Article  MATH  Google 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.

    MathSciNet  Article  MATH  Google Scholar 

  24. [24]

    A. Rinot: Chain conditions of products, and weakly compact cardinals, Bull. Symb. Log. 20 (2014), 293–314.

    Article  MATH  Google Scholar 

  25. [25]

    A. Rinot: The Ostaszewski square, and homogeneous Souslin trees, Israel J. Math. 199 (2014), 975–1012.

    MathSciNet  Article  MATH  Google Scholar 

  26. [26]

    A. Rinot: Chromatic numbers of graphs - large gaps, Combinatorica 35 (2015), 215–233.

    MathSciNet  Article  MATH  Google Scholar 

  27. [27]

    A. Rinot: Hedetniemi’s conjecture for uncountable graphs, J. Eur. Math. Soc. (JEMS) 19 (2017), 285–298.

    MathSciNet  Article  MATH  Google Scholar 

  28. [28]

    A. Rinot: Higher Souslin trees and the GCH, revisited, Adv. Math. 311 (2017), 510–531.

    MathSciNet  Article  MATH  Google Scholar 

  29. [29]

    A. Rinot: Same graph, different universe, Arch. Math. Logic, 56 (2017), 783–796.

    MathSciNet  Article  MATH  Google 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.

    MathSciNet  Article  MATH  Google Scholar 

  32. [32]

    S. Shelah: Incompactness for chromatic numbers of graphs, in: A tribute to Paul Erdős, 361–371, Cambridge Univ. Press, Cambridge, 1990.

    Google Scholar 

  33. [33]

    S. Shelah: Re ecting stationary sets and successors of singular cardinals, Arch. Math. Logic 31 (1991), 25–53.

    MathSciNet  Article  MATH  Google Scholar 

  34. [34]

    S. Shelah: On incompactness for chromatic number of graphs, Acta Mathematica Hungarica 139 (2013), 363–371.

    MathSciNet  Article  MATH  Google Scholar 

  35. [35]

    L. Soukup: On c+-chromatic graphs with small bounded subgraphs, Period. Math. Hungar. 21 (1990), 1–7.

    MathSciNet  Article  MATH  Google Scholar 

  36. [36]

    S. Todorčević: On a conjecture of R. Rado, J. London Math. Soc. (2) 27 (1983), 1–8.

    MathSciNet  MATH  Google 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 

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Correspondence to Chris Lambie-Hanson.

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Lambie-Hanson, C., Rinot, A. Reflection on the Coloring and Chromatic Numbers. Combinatorica 39, 165–214 (2019). https://doi.org/10.1007/s00493-017-3741-6

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Mathematics Subject Classification (2010)

  • 03E35
  • 05C15
  • 05C63