, Volume 35, Issue 2, pp 215–233 | Cite as

Chromatic numbers of graphs — large gaps

  • Assaf RinotEmail author
Original Paper


We say that a graph G is (ℵ0,κ)-chromatic if Chr(G) = κ, while Chr(G′) ≤ ℵ0 for any subgraph G′ of G of size < |G|.

The main result of this paper reads as follows. If □λ+CHλ holds for a given uncountable cardinal λ, then for every cardinal κ≤λ, there exists an (ℵ0,κ)-chromatic graph of size λ+.

We also study (ℵ0+)-chromatic graphs of size λ+. In particular, it is proved that if 0 # does not exist, then for every singular strong limit cardinal λ, there exists an (ℵ0+)-chromatic graph of size λ+.

Mathematics Subject Classification (2010)

03E35 05C15 05C63 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    James E. Baumgartner: Generic graph construction, J. Symbolic Logic 49 (1984), 234–240.CrossRefzbMATHMathSciNetGoogle Scholar
  2. [2]
    N. G. de Bruijn and P. Erdos: A colour problem for infinite graphs and a problem in the theory of relations, Nederl. Akad. Wetensch. Proc. Ser. A. 54; Indagationes Math. 13 (1951), 369–373.Google Scholar
  3. [3]
    P. Erdos and A. Hajnal: On chromatic number of graphs and set-systems, Acta Math. Acad. Sci. Hungar 17 (1966), 61–99.CrossRefMathSciNetGoogle Scholar
  4. [4]
    P. Erdos 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
  5. [5]
    M. Foreman and Richard Laver: Some downwards transfer properties for ℵ2, Adv. in Math. 67 (1988), 230–238.CrossRefzbMATHMathSciNetGoogle Scholar
  6. [6]
    F. Galvin: Chromatic numbers of subgraphs, Period. Math. Hungar. 4 (1973), 117–119.CrossRefzbMATHMathSciNetGoogle Scholar
  7. [7]
    P. Komjáth: Consistency results on infinite graphs, Israel J. Math. 61 (1988), 285–294.CrossRefzbMATHMathSciNetGoogle Scholar
  8. [8]
    P. Komjáth: Subgraph chromatic number, in: Set theory (Piscataway, NJ, 1999), vol. 58 of DIMACS Ser. Discrete Math. Theoret. Comput. Sci., 99–106. Amer. Math. Soc., Providence, RI, 2002.Google Scholar
  9. [9]
    W. J. Mitchell: The covering lemma, in: Handbook of set theory. Vols. 1, 2, 3, 1497–1594. Springer, Dordrecht, 2010.CrossRefGoogle Scholar
  10. [10]
    A. Rinot: The Ostaszewski square, and homogeneous Souslin trees, Israel J. Math. 199 (2014), 975–1012.CrossRefzbMATHMathSciNetGoogle Scholar
  11. [11]
    A. Rinot: Jensen’s diamond principle and its relatives, in: Set theory and its applications, volume 533 of Contemp. Math., 125–156. Amer. Math. Soc., Providence, RI, 2001.Google Scholar
  12. [12]
    S. Shelah: On incompactness for chromatic number of graphs, Acta Mathematica Hungarica, accepted.Google Scholar
  13. [13]
    S. Shelah: Incompactness for chromatic numbers of graphs, in: A tribute to Paul Erdos, 361–371. Cambridge Univ. Press, Cambridge, 1990.CrossRefGoogle Scholar
  14. [14]
    L. Soukup: On c+-chromatic graphs with small bounded subgraphs, Period. Math. Hungar. 21 (1990), 1–7.CrossRefzbMATHMathSciNetGoogle Scholar
  15. [15]
    S. Todorčević: Comparing the continuum with the first two uncountable cardinals, in: Logic and scientific methods (Florence, 1995), vol. 259 of Synthese Lib., 145–155. Kluwer Acad. Publ., Dordrecht, 1997.Google Scholar
  16. [16]
    S. Todorcevic: Combinatorial dichotomies in set theory, Bull. Symbolic Logic 17 (2011), 1–72.CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© János Bolyai Mathematical Society and Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.The Fields Institute for Research in Mathematical SciencesTorontoCanada
  2. 2.Department of MathematicsUniversity of TorontoTorontoCanada

Personalised recommendations