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⩽1/2−1/2n, and the edges of K⌊(2−β)n⌋ be 2-colored so that no monochromatic Cn 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 Kq,⌊(2−β)n⌋−q−1, while the other one induces Kq∪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 for every 2-coloring of E(G) one of the colors contains cycles Ct for all t∈[3,n].