On powers of Hamilton cycles in Ramsey-Turán Theory

We prove that for $r\in \mathbb{N}$ with $r\ge 2$ and $\mu >0$, there exist $\alpha >0$ and $n_0$ such that for every $n\ge n_0$, every $n$-vertex graph $G$ with $\delta (G)\ge (1-\frac{1}{r}+\mu )n$ and $\alpha (G)\le \alpha n$ contains an $r$-th power of a Hamilton cycle. We also show that the minimum degree condition is asymptotically sharp for $r=2,3$ and the $r=2$ case was recently conjectured by Staden and Treglown.