Set colourings of graphs

An r-set colouring of a graph G is an assignment of r distinct colours to each vertex of G   so that the sets of colours assigned to adjacent vertices are disjoint. We denote by χ(r)(G)χ(r)(G) the minimum number of colours required to r-set colour G. The set-chromatic number of G  , denoted by χ*(G)χ*(G), is defined byχ*(G)=infrχ(r)(G)r.Clearly 2⩽χ*(G)⩽χ(G)2⩽χ*(G)⩽χ(G).By making use of a recent result of L. Lovász we prove that min{χ(r)(G):χ(G)=k}=2r+k-2andmin{χ(r)(G):Gisuniquelyk-colourable}=2r+k-1. In particular, given any k,χ*(G)k,χ*(G) may be arbitrarily close to 2, and given any r  , the ratio χ(r)(H)/χ(H)χ(r)(H)/χ(H) may be arbitrarily close to 1, even if H is uniquely colourable. The results disprove a conjecture of D.P. Geller.