Andrasfai and Vega graphs in Ramsey-Turan theory

Given positive integers n > s, we let ex ( n , s ) denote the maximum number of edges in a triangle-free graph G on n vertices with alpha ( G ) <= s. In the early 1960s, Andrasfai conjectured that for n / 3 < s < n / 2 the function ex ( n , s ) is piecewise quadratic with critical values at s / n = k / ( 3 k - 1 ) for k is an element of N. We confirm that this is indeed the case whenever s / n is slightly larger than a critical value, thus determining ex ( n , s ) for all n and s such that s / n is an element of [ k / ( 3 k - 1 ) , k / ( 3 k - 1 ) + gamma k ], where gamma k = Theta ( k - 6 ).