One of the earliest results in extremal graph theory, Mantel’s theorem, states that the maximum number of edges in a triangle-free graph G on n vertices is n2/4c. We investigate how this extremal bound is affected when G is additionally required to contain a prescribed graph H as a subgraph. We establish general upper and lower bounds for this problem, which are tight in the exponent for random trianglefree graphs and graphs generated by the triangle-free process, when the size of H lies within certain ranges.