On The Restricted Size Ramsey Number

Let F, G, and H be simple graphs. We say F → (G, H) if for every 2-coloring of the edges of F there exists a monochromatic G or H in F. The Ramsey number r(G, H) is defined as min {|V (F)|: F → (G, H)}, while the restricted size Ramsey number r∗(G, H) is defined as min {|E (F)|: F → (G, H), |V (F) | = r(G, H)}. In this paper, we give lower and upper bounds for the restricted size Ramsey number for a path P3 versus cycles Cn