Long rainbow cycles in proper edge-colorings of complete graphs

We show that any properly edge-colored Kn contains a rainbow cycle with at least (4=7 − o(1))n edges. This improves the lower bound of n=2 − 1 proved in [1]

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