Some infinite families of Ramsey ($${varvec{P}}_mathbf{3},{varvec{P}}_{{varvec{n}}}$$)-minimal trees

For any given two graphs G and H, the notation (Frightarrow ) (G, H) means that for any red–blue coloring of all the edges of F will create either a red subgraph isomorphic to G or a blue subgraph isomorphic to H. A graph F is a Ramsey (G, H)-minimal graph if (Frightarrow ) (G, H) but (F-enrightarrow (G,H)), for every (e in E(F)). The class of all Ramsey (G, H)-minimal graphs is denoted by (mathcal {R}(G,H)). In this paper, we construct some infinite families of trees belonging to (mathcal {R}(P_3,P_n)), for (n=8) and 9. In particular, we give an algorithm to obtain an infinite family of trees belonging to (mathcal {R}(P_3,P_n)), for (nge 10).