Edge partitions of the Rado graph

Abstract

We will prove that for every colouring of the edges of the Rado graph,ℛ (that is the countable homogeneous graph), with finitely many coulours, it contains an isomorphic copy whose edges are coloured with at most two of the colours. It was known [4] that there need not be a copy whose edges are coloured with only one of the colours. For the proof we use the lexicographical order on the vertices of the Rado graph, defined by Erdös, Hajnal and Pósa.

Using the result we are able to describe a “Ramsey basis” for the class of Rado graphs whose edges are coloured with at most a finite number,r, of colours. This answers an old question of M. Pouzet.

This is a preview of subscription content, access via your institution.

References

  1. [1]

    R. Fraïssé: Theory of Relations,Studies in Logic and the Foundations of Mathematics, Elsevier Science Publishing Co., Inc., U.S.A.,118 (1986), 313.

    Google Scholar 

  2. [2]

    S. Todorcevic: Some partitions of three-dimensional combinatorial cubes,Journal of Combinatorial Theory A,68 (1994), 410–437.

    Google Scholar 

  3. [3]

    D. Devlin:Some partition theorems and ultrafilters on ω, Ph. D. Thesis, Dartmouth College, 1979.

  4. [4]

    P. Erdős, A. Hajnal, L. Pósa: Strong embeddings of graphs into coloured graphs,Colloquia Mathematica Societatis János Bolyai,10, Vol. I, Infinite and finite sets, 585–595, Edited by A. Hajnal, R. Rado, Vera T. Sós.

Download references

Author information

Affiliations

Authors

Additional information

Supported by the French PRC Math-Info.

Supported by NSERC of Canada Grant # 691325.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Pouzet, M., Sauer, N. Edge partitions of the Rado graph. Combinatorica 16, 505–520 (1996). https://doi.org/10.1007/BF01271269

Download citation

Mathematics Subject Classification (1991)

  • 05 C 55
  • 04 A 20