For graphs Fnβ and Gnβ of order n, if R(Fnβ,Gnβ)=(Ο(Gnβ)β1)(nβ1)+Ο(Gnβ), then Fnβ is said to be Gnβ-good,
where Ο(Gnβ) is the minimum size of a color class among all proper
vertex-colorings of Gnβ with Ο(Gnβ) colors. Given Ξ(Gnβ)β€Ξ, it is shown that Pnβ is asymptotically Gnβ-good if
Ξ±(Gnβ)β€4nβ.Comment: 8 pages, 3 figure