Andrásfai and Vega graphs in Ramsey–Turán theory

Given positive integers (Formula presented.), we let (Formula presented.) denote the maximum number of edges in a triangle-free graph (Formula presented.) on (Formula presented.) vertices with (Formula presented.). In the early 1960s, Andrásfai conjectured that for (Formula presented.) the function (Formula presented.) is piecewise quadratic with critical values at (Formula presented.) for (Formula presented.). We confirm that this is indeed the case whenever (Formula presented.) is slightly larger than a critical value, thus determining (Formula presented.) for all (Formula presented.) and (Formula presented.) such that (Formula presented.), where (Formula presented.).