Ramsey numbers involving a long path

Let HH be a graph, and σ(H)σ(H) the minimum size of a color class among all proper vertex-coloring of HH with χ(H)χ(H) colors. A connected graph GG with |G|≥σ(H)|G|≥σ(H) is said to be HH-good if R(G,H)=(χ(H)−1)(|G|−1)+σ(H)R(G,H)=(χ(H)−1)(|G|−1)+σ(H). In this note, we shall show that if n≥8|H|+3σ2(H)+cχ8(H)n≥8|H|+3σ2(H)+cχ8(H), then PnPn is HH-good, where c=1014c=1014.