Restricted size Ramsey number of disjoint union of stars versus a complete graph

Restricted size Ramsey number of disjoint union of stars versus a complete graph

For given simple graphs F,G and H, we write F→(G,H) if in every 2-coloring of the edges of F there exists a monochromatic copy of G or H. The Ramsey number R(G,H) is defined as the smallest positive integer n such that K_n→(G,H). The restricted size Ramsey number r^* (G,H) is defined as the min{|E(F)| ∶F → (G,H),|V (F)| = R(G,H)}. In this note, the exact value of the restricted size Ramsey number of disjoint copies of stars versus a complete graph is determined.

For given simple graphs F,G and H, we write F→(G,H) if in every 2-coloring of the edges of F there exists a monochromatic copy of G or H. The Ramsey number R(G,H) is defined as the smallest positive integer n such that K_n→(G,H). The restricted size Ramsey number r^* (G,H) is defined as the min{|E(F)| ∶F → (G,H),|V (F)| = R(G,H)}. In this note, the exact value of the restricted size Ramsey number of disjoint copies of stars versus a complete graph is determined.