The next case of Andrásfai's conjecture

Let $\mathrm{e}\mathrm{x}(n,s)$ denote the maximum number of edges in a triangle-free graph on $n$ vertices which contains no independent sets larger than $s$. The behaviour of $\mathrm{e}\mathrm{x}(n,s)$ was first studied by Andrásfai, who conjectured that for $s>n\mathrm{/}3$ this function is determined by appropriately chosen blow-ups of so called Andrásfai graphs. Moreover, he proved $\mathrm{e}\mathrm{x}(n,s)=n^2-4ns+5s^2$ for $s\mathrm{/}n\in [2\mathrm{/}5,1\mathrm{/}2]$ and in earlier work we obtained $\mathrm{e}\mathrm{x}(n,s)=3n^2-15ns+20s^2$ for $s\mathrm{/}n\in [3\mathrm{/}8,2\mathrm{/}5]$. Here we make the next step in the quest to settle Andrásfai’s conjecture by proving $\mathrm{e}\mathrm{x}(n,s)=6n^2-32ns+44s^2$ for $s\mathrm{/}n\in [4\mathrm{/}11,3\mathrm{/}8]$.