One of the oldest results in modern graph theory, due to Mantel, asserts that
every triangle-free graphs on n vertices has at most ⌊n2/4⌋
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 αn, where 2/5≤α<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 α(G)≥3n/8 and state a conjecture on the structure of
the densest triangle-free graphs G with α(G)>n/3. We remark that the
case α(G)≤n/3 behaves differently, but due to the work of Brandt
this situation is fairly well understood.Comment: Revised according to referee report