I am trying to show that $ \forall \alpha >0, \exists C(\alpha)$:
If $G$ is an $n$-vertex graph with minimum degree at least $\alpha n$, then either $G$ contains $C_7$ as a subgraph, or it is $C(\alpha)$-colourable.
My attempt until now, includes the corresponding proof of Thomassen for $C_5$ instead, along with the proof of Tomasz Łuczak and Stéphan Thomassé but I do not see a possible way into extending the result to $C_7$.