Low independence number and Hamiltonicity implies pancyclicity

A graph on $n$ vertices is called pancyclic if it contains a cycle of every length $3\le l \le n$. Given a Hamiltonian graph $G$ with independence number at most $k$ we are looking for the minimum number of vertices $f(k)$ that guarantees that $G$ is pancyclic. The problem of finding $f(k)$ was raised by Erd\H{o}s in 1972 who showed that $f(k)\le 4k^4$, and conjectured that $f(k)=\Theta(k^2)$. Improving on a result of Lee and Sudakov we show that $f(k)=O(k^{11/5})$.