Fix a target graph H and a family *F* of forbidden graphs. The *generalized extremal number *ex(n, H, *F*) is the maximum number of H-copies possible in an n-vertex graph which avoids *F*. Note that when H is an edge, ex(n, H, *F*) is the ordinary extremal number ex(n, *F*). After the systematic study of generalized extremal numbers was initiated by Alon and Shikhelman in 2016, the area has received substantial attention. In addition to explicit computation of ex(n, H, *F*) for specific choices of H, *F*, many questions in extremal graph theory (e.g., supersaturation, stability) naturally extend to the generalized setting.

In 2015, Bukh and Conlon applied the random algebraic method to show that, for any rational r in the interval [1,2], there is a family *F *such that ex(n, *F*) = Theta(n^r). Analogously, for a fixed target graph H and a rational number r within an appropriate interval, we may ask whether it is possible to find a forbidden family *F *for which ex(n, H, *F*) = Theta(n^r). In this talk, we present results on this question for some specific target graphs H, focusing on the case where H is a triangle, for which we show that all rational exponents in [1,3] are realizable. Joint work with Sean English and Bob Krueger.