Let $k\ge 0$. Let $f(k)$ be such that every graph on $n\ge f(k)$ vertices with at least $\left(\genfrac{}{}{0ex}{}{n-k-1}{2}\right)+\left(\genfrac{}{}{0ex}{}{k+2}{2}\right)+1$ edges contains a cycle on $n-k$ vertices. Determine or estimate $f(k)$.