Hamiltonicity, independence number, and pancyclicity

A graph on n vertices is called pancyclic if it contains a cycle of length ? for all 3≤?≤n. In 1972, Erd?s proved that if G is a Hamiltonian graph on n>4k⁴ vertices with independence number k, then G is pancyclic. He then suggested that n=Ω(k²) should already be enough to guarantee pancyclicity. Improving on his and some other later results, we prove that there exists a constant c such that n>ck^7/3 suffices.