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.