Small graphs on Ramsey minimal P₄ versus P₆ Available to Purchase

We consider two simple graphs G and H, the notation F → (G, H) means that for any red- blue coloring of all the edges of graph F contains either a red copy isomorphic to G or a blue copy isomorphic to H. A graph F is Ramsey (G, H)-minimal graph if F → (G, H) and for any edge e in F then F - e → (G, H). The set of all Ramsey minimal graphs for pair (G, H) is denoted by R(G, H). The Ramsey set for pair (G, H) is said to be Ramsey-finite or Ramsey-infinite if R(G, H) is finite or infinite, respectively. Several articles have discussed the problem of determining whether R(G, H) is finite (infinite). It is known that the set R(P_m, P_n), for 3 ≤ m ≤ n is Ramsey-infinite. Some partial results in R(P₄, P_n), for n = 4 and n = 5 , have been obtained. Motivated by this, we are interested in determining graphs in R(P₄, P₆). In this paper, we determine some graphs of certain order in R(P₄, P₆).