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 R_ (G, H) is the largest n such that any red-blue edge coloring of Kᵣ P₍ contains a red G or a blue H, where r=R (G, H) and P₍ is a path of order n. In this note, we show a general upper bound for R_ (G, H), and determine the exact values for some cases of R_ (G, H).