Israel Journal of Mathematics

, Volume 61, Issue 3, pp 285–294 | Cite as

Consistency results on infinite graphs

  • Péter Komjáth


Consistently there exist ℵ2-chromatic graphs with no ℵ1-chromatic subgraphs. The statement that every uncountably chromatic graph of size ℵ1 contains an uncountably chromaticω-connected subgraph is consistent and independent. It is consistent that there is an uncountably chromatic graph of size ℵω 1 in which every subgraph with size less than ℵω 1 is countably chromatic.


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  1. 1.
    J. E. Baumgartner,Generic graph construction, J. Symb. Logic49 (1984), 234–240.MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    W. W. Comfort and S. Negrepontis,Chain Conditions in Topology, Cambridge University Press, 1982.Google Scholar
  3. 3.
    P. Erdös,Problems and results on finite and infinite combinatorial analysis, inInfinite and Finite Sets, Coll. Math. Soc. J. Bolyai, 10 (A. Hajnal, R. Rado and V. T. Sós, eds.), pp. 403–424.Google Scholar
  4. 4.
    P. Erdös,Problems and results on finite and infinite combinatorial analysis, II, Enseign. Math.27 (1981), 163–176.MATHMathSciNetGoogle Scholar
  5. 5.
    P. Erdös,On the combinatorial problems I would most like to see solved, Combinatorica1 (1981), 25–42.MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    P. Erdös,Problems and results on chromatic numbers in finite and infinite graphs, inGraph Theory with Applications to Algorithms and Computer Science (Y. Alavi, G. Chartrand, L. Lesniak, D. R. Lich and C. E. Wall, eds.), John Wiley and Sons, 1985, pp. 201–213.Google Scholar
  7. 7.
    P. Erdös and A. Hajnal,On chromatic number of graphs and set-systems, Acta Math. Acad. Sci. Hung.17 (1966), 61–99.MATHCrossRefGoogle Scholar
  8. 8.
    P. Erdös and A. Hajnal,Chromatic number of finite and infinite graphs and hypergraphs, Discr. Math.53 (1985), 281–285.MATHCrossRefGoogle Scholar
  9. 9.
    M. Foreman,Potent axioms, Trans Am. Math. Soc.294 (1986), 1–28.MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    M. Foreman and R. Laver,A graph transfer property, to appear.Google Scholar
  11. 11.
    F. Galvin,Chromatic numbers of subgraphs, Period. Math. Hung.4 (1973), 117–119.MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    P. Komjáth,Connectivity and chromatic number of infinite graphs, Isr. J. Math.56 (1986), 257–266.MATHGoogle Scholar
  13. 13.
    P. Komjáth and S. Shelah,Forcing constructions for uncountable chromatic graphs, J. Symb. Logic, to appear.Google Scholar
  14. 14.
    S. Shelah,Remarks on cardinal invariants in topology, Gen. Topol. Appl.7 (1977), 251–259.Google Scholar
  15. 15.
    S. Shelah,A compactness theorem for singular cardinals, free algebras, Whitehead problem, and transversals, Isr. J. Math.21 (1975), 319–349.MATHCrossRefGoogle Scholar
  16. 16.
    S. Shelah,Proper Forcing, Lecture Notes in Math.940, Springer-Verlag, Berlin, 1982.MATHGoogle Scholar

Copyright information

© Hebrew University 1988

Authors and Affiliations

  • Péter Komjáth
    • 1
    • 2
  1. 1.Department of MathematicsLehigh UniversityBethlehemUSA
  2. 2.Department of Computer ScienceEötvös UniversityBudapestHungary

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