Andrásfai and Vega graphs in Ramsey–Turán theory

Given positive integers $n\ge s$, we let ${\mathrm{ex}}(n,s)$ denote the maximum number of edges in a triangle-free graph $G$ on $n$ vertices with $\alpha(G)\le s$. In early sixties Andr\'{a}sfai conjectured that for $n/3<s<n/2$ function ${\mathrm{ex}}(n, s)$ is piecewise quadratic critical values at $s/n={k}/({3k-1})$. We confirm this indeed case whenever $s/n$ slightly larger than value, thus determining all and $s$ such $s/n\in [{k}/({3k-1}), {k}/({3k-1})+\gamma_k]$, where $\gamma_k=\Theta(k^{-6})$.