On some path-critical Ramsey numbers

For graphs G and H, the Ramsey number $R(G,H)$ is the smallest r such that any red-blue edge coloring of Kᵣ contains a red G or a blue H. The path-critical Ramsey number Rπ(G,H) is the largest n such that any red-blue edge coloring of Kᵣ Pₙ contains a red G or a blue H, where $r=R(G,H)$ and Pₙ is a path of order n. In this note, we show a general upper bound for Rπ(G,H), and determine the exact values for some cases of Rπ(G,H).