Cycles and stability

Abstract

We prove a number of Turán and Ramsey type stability results for cycles, in particular, the following one: Let $n>4$, $0<\beta ⩽1/2-1/2n$, and the edges of $K_{\lfloor (2-\beta )n\rfloor }$ be 2-colored so that no monochromatic $C_n$ exists. Then, for some $q\in ((1-\beta )n-1,n)$, we may drop a vertex v so that in $K_{\lfloor (2-\beta )n\rfloor }-v$ one of the colors induces $K_{q,\lfloor (2-\beta )n\rfloor -q-1}$, while the other one induces $K_q\cup K_{\lfloor (2-\beta )n\rfloor -q-1}$. 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 $\delta (G)⩾(2-{10}^{\mathrm{-6}})n$, then for every 2-coloring of $E(G)$ one of the colors contains cycles $C_t$ for all $t\in [3,n]$.