On Graphs With Small Ramsey Numbers, II

For arbitrary graphs G1 and G2, define the Ramsey number R(G1,G2) to be the minimum positive integer N such that in every bicoloring of edges of the complete graph KN with, say, red and blue colors, there is either a red copy of G1 or a blue copy of G2. The classical Ramsey number r(k, l) is in our terminology R(Kk,Kl). Call a family F of graphs linear Ramsey if there exists a constant C = C(F) such that for every G∈F , R(G,G) ≤ C|V (G)|.