Locally restricted colorings

We consider the following problem: given suitable integers χχ and p  , what is the smallest value ρρ such that, for any graph G   with chromatic number χχ and any vertex coloring of G   with at most χ+pχ+p colors, there is a vertex vv such that at least χχ different colors occur within distance ρρ of vv? Let ρ(χ,p)ρ(χ,p) be this value; we show in particular that ρ(χ,p)⩽⌈p/2⌉+1ρ(χ,p)⩽⌈p/2⌉+1 for all χ,pχ,p. We give the exact value of ρρ when p=0p=0 or χ⩽3χ⩽3, and (χ,p)=(4,1)(χ,p)=(4,1) or (4,2)(4,2).