Restricted size Ramsey number for P3 versus cycle - Joanna Cyman; Tomasz Dzido (GTA Research Group, Univ. Newcastle, Indonesian Combinatorics Society and ITB, 2020)
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 red copy of G or a blue copy of H in F. The Ramsey number r(G,H) is defined as r(G,H) = min|V(F)|: F → (G,H), while the restricted size Ramsey number r*(G,H) is defined as r*(G,H) = min|E(F)|: F → (G,H),|V(F)| = r(G,H). In this paper we determine previously unknown restricted size Ramsey numbers r*(P3,Cn) for 7 ≤ n ≤ 12. We also give new upper bound r*(P3,Cn) ≤ 2n-2 for n ≥ 10 and n is even.