Cycles and stability

We prove a number of Turán and Ramsey type stability results for cycles, in particular, the following one: Let n > 4 , 0 < β ⩽ 1 / 2 − 1 / 2 n , and the edges of K ⌊ ( 2 − β ) n ⌋ be 2-colored so that no monochromatic C n exists. Then, for some q ∈ ( ( 1 − β ) n − 1 , 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 2 n − 1 , with minimum degree δ ( G ) ⩾ ( 2 − 10 −6 ) n , then for every 2-coloring of E ( G ) one of the colors contains cycles C t for all t ∈ [ 3 , n ] .