Tur\'an-type problems for long cycles in random and pseudo-random graphs

We study the Tur\'an number of long cycles in random graphs and in pseudo-random graphs. Denote by $ex(G(n,p),H)$ the random variable counting the number of edges in a largest subgraph of $G(n,p)$ without a copy of $H$. We determine the asymptotic value of $ex(G(n,p), C_t)$ where $C_t$ is a cycle of length $t$, for $p\geq \frac Cn$ and $A \log n \leq t \leq (1 - \varepsilon)n$. The typical behavior of $ex(G(n,p), C_t)$ depends substantially on the parity of $t$. In particular, our results match the classical result of Woodall on the Tur\'an number of long cycles, and can be seen as its random version, showing that the transference principle holds here as well. In fact, our techniques apply in a more general sparse pseudo-random setting. We also prove a robustness-type result, showing the likely existence of cycles of prescribed lengths in a random subgraph of a graph with a nearly optimal density.