Cycles and stability

We prove a number of Turán and Ramsey type stability results for cycles, in particular, the following one: Let n be sufficiently large, 0 < β < 2 × 10 −7 , and the edges of K ⌊(2−β)n⌋ be 2-colored so that no mononchromatic C n exists. Then, for some q ∈ ((1 − β) n, n) , we may drop a vertex v so that in K ⌊(2−β)n⌋ −v one of the colors induces K q,⌊(2−β)n⌋−q−1 , while the other one induces K q ∪ K ⌊(2−β)n⌋−q−1. We also derive the following Ramsey type result. If n is sufficiently large and G is a graph of order 2n − 1, with minimum degree δ (G) ≥ 2 − 10 −6 n, then every 2-coloring of E (G) produces a mononchromatic cycle C n .