Sparse Approximate Chromatic Profiles of Triangle-Free Graphs

We prove a sparse version of the four-colour theorem of Brandt and Thomassé, answering a question of Allen, Böttcher, Kohayakawa and Roberts. For every fixed $0<γ\le1/10$ and every $p=p(n)\in(0,1]$, asymptotically almost surely every spanning triangle-free $H\subseteq G(n,p)$ with $δ(H)\ge(1/3+γ)pn$