Extremal triangle-free and odd-cycle-free colourings of uncountable graphs


The optimality of the Erdős–Rado theorem for pairs is witnessed by the colouring \(\Delta_\kappa : [2^\kappa]^2 \rightarrow \kappa\) recording the least point of disagreement between two functions. This colouring has no monochromatic triangles or, more generally, odd cycles. We investigate a number of questions investigating the extent to which \(\Delta_\kappa\) is an extremal such triangle-free or odd-cycle-free colouring. We begin by introducing the notion of \(\Delta\)-regressive and almost \(\Delta\)-regressive colourings and studying the structures that must appear as monochromatic subgraphs for such colourings. We also consider the question as to whether \(\Delta_\kappa\) has the minimal cardinality of any maximal triangle-free or odd-cycle-free colouring into \(\kappa\). We resolve the question positively for odd-cycle-free colourings.

This is a preview of subscription content, log in to check access.


  1. 1.

    J. E. Baumgartner, Ineffability properties of cardinals. I, in: Infinite and Finite Sets (Colloq., Keszthely, 1973; dedicated to P. Erdős on his 60th birthday), Vol. I, Colloq. Math. Soc. János Bolyai, Vol. 10, North-Holland (Amsterdam, 1975), pp. 109–130

  2. 2.

    Erdős, P., Rado, R.: A construction of graphs without triangles having pre-assigned order and chromatic number. J. London Math. Soc. 35, 445–448 (1960)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Erdös, P., Rado, R.: A partition calculus in set theory. Bull. Amer. Math. Soc. 62, 427–489 (1956)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Sy-David Friedman, Tapani Hyttinen, and Vadim Kulikov, Generalized descriptive set theory and classification theory, Mem. Amer. Math. Soc., 230 (2014), no. 1081

  5. 5.

    Hajnal, András, Komjáth, Péter: Some remarks on the simultaneous chromatic number. Combinatorica 23, 89–104 (2003)

    MathSciNet  Article  Google Scholar 

  6. 6.

    András Hajnal and Attila Máté, Set mappings, partitions, and chromatic numbers, in: Logic Colloquium '73 (Bristol, 1973), North-Holland (Amsterdam, 1975), pp. 347–379

  7. 7.

    Komjáth, Péter: A note on Hajnal-Máté graphs. Studia Sci. Math. Hungar. 15, 275–276 (1980)

    MathSciNet  MATH  Google Scholar 

  8. 8.

    Komjáth, Péter: A second note on Hajnal-Máté graphs. Studia Sci. Math. Hungar. 19, 245–246 (1984)

    MathSciNet  MATH  Google Scholar 

  9. 9.

    Komjáth, Péter, Shelah, Saharon: Forcing constructions for uncountably chromatic graphs. J. Symbolic Logic 53, 696–707 (1988)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Rödl, V.: On the chromatic number of subgraphs of a given graph. Proc. Amer. Math. Soc. 64, 370–371 (1977)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Todorčević, Stevo: Partitioning pairs of countable ordinals. Acta Math. 159, 261–294 (1987)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Todorčević, Stevo: Some partitions of three-dimensional combinatorial cubes. J. Combin. Theory Ser. A 68, 410–437 (1994)

    MathSciNet  Article  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to C. Lambie-Hanson.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Lambie-Hanson, C., Soukup, D.T. Extremal triangle-free and odd-cycle-free colourings of uncountable graphs. Acta Math. Hungar. (2020). https://doi.org/10.1007/s10474-020-01053-2

Download citation

Key words and phrases

  • Ramsey theory
  • regressive colouring
  • triangle-free colouring
  • uncountable graph

Mathematics Subject Classification

  • primary 03E02
  • secondary 05C63
  • 03E05