On the Ramsey-Turán Density of Triangles

One of the oldest results in modern graph theory, due to Mantel, asserts that every triangle-free graph on n vertices has at most ⌊ n 2 /4⌋ edges. About half a century later Andrásfai 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 Andrásfai’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 ) ≥ 3 n /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 ) α n /3 behaves differently, but due to the work of Brandt this situation is fairly well understood.