Ramsey minimal graphs
As usual, for graphs Γ, G, and H, we write Γ → (G, H) to mean that any red-blue colouring of the edges of Γ contains a red copy of G or a blue copy of H. A pair of graphs (G, H) is said to be Ramsey-infinite if there are infinitely many minimal graphs Γ for which we have Γ → (G, H). Let ℓ ≥ 4 be an integer. We show that if H is a 2-connected graph that does not contain induced cycles of length at least ℓ, then the pair (Ck, H) is Ramsey-infinite for any k ≥ ℓ, where Ck denotes the cycle of length k.
Journal of the Brazilian Computer Society
Bollobás, B., Donadelli, J., Kohayakawa, Y., & Schelp, R. (2001). Ramsey minimal graphs. Journal of the Brazilian Computer Society, 7 (3), 27-37. https://doi.org/10.1590/S0104-65002001000200005