Some Remarks on the Simultaneous Chromatic Number

We present several partial results, variants, and consistency results concerning the following (as yet unsolved) conjecture. If X is a graph on the ground set V with \( Chr{\left( X \right)} = {\aleph }_{1} \) then X has an edge coloring F with \( {\aleph }_{1} \) colors such that if V is decomposed into \( {\aleph }_{0} \) parts then there is one in which F assumes all values.

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Correspondence to András Hajnal*.

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Due to some unfortunate misunderstandings, this paper appeared much later than we expected.

* Research partially supported by NSF grants DMS-9704477 and DMS-0072560.

† Research partially supported by Hungarian National Research Grant T 032455.

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Hajnal*, A., Komjáth†, P. Some Remarks on the Simultaneous Chromatic Number. Combinatorica 23, 89–104 (2003).

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AMS Subject Classification (2000):

  • 03E35
  • 05C15