On the Ramsey-Tur\'an density of triangles

One of the oldest results in modern graph theory, due to Mantel, asserts that every triangle-free graphs on $n$ vertices has at most $\lfloor n^2/4\rfloor$ edges. About half a century later Andr\'asfai studied dense triangle-free graphs and proved that the largest triangle-free graphs on $n$ vertices without independent sets of size $\alpha n$, where $2/5\le \alpha < 1/2$, are blow-ups of the pentagon. More than 50 further years have elapsed since Andr\'asfai's work. In this article we make the next step towards understanding the structure of dense triangle-free graphs without large independent sets. Notably, we determine the maximum size of triangle-free graphs~$G$ on $n$ vertices with $\alpha (G)\ge 3n/8$ and state a conjecture on the structure of the densest triangle-free graphs $G$ with $\alpha(G) > n/3$. We remark that the case $\alpha(G) \le n/3$ behaves differently, but due to the work of Brandt this situation is fairly well understood.