Andrásfai and Vega graphs in Ramsey–Turán 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 α ( G ) ⩽ s. In the early 1960s, Andrásfai 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 ∈ 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 ∈ [ k ∕ ( 3 k − 1 ) , k ∕ ( 3 k − 1 ) + γ k ], where γ k = Θ ( k − 6 ).