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  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.
Mathematics Subject Classification (1991)05 C 55 04 A 20
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- 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
- S. Todorcevic: Some partitions of three-dimensional combinatorial cubes,Journal of Combinatorial Theory A,68 (1994), 410–437.Google Scholar
- D. Devlin:Some partition theorems and ultrafilters on ω, Ph. D. Thesis, Dartmouth College, 1979.Google Scholar
- 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.Google Scholar