Polynomial minimum degree stability for $3$-chromatic graphs

Let $H$ be a fixed 3-chromatic graph, and let $g \geq 2$ be the smallest integer such that there is no homomorphism $H \rightarrow C_{2 g+1}$. For every $\varepsilon>0$, we prove that there exists a constant $ρ>0$ such that every $H$-free $n$-vertex graph $G$ with $δ(G) \geq(2 /(2 g+1)+\varepsilon)