  Loh, Tait, Timmons and Zhou introduced the notion of induced Tur\'{a}n
numbers, defining $\operatorname{ex}(n, \{H, F\text{-ind}\})$ to be the
greatest number of edges in an $n$-vertex graph with no copy of $H$ and no
induced copy of $F$. Their and subsequent work has focussed on $F$ being a
complete bipartite graph. In this short note, we complement this focus by
asymptotically determining the induced Tur\'{a}n number whenever $H$ is not
bipartite and $F$ is not an independent set nor a complete bipartite graph.

Combinatorics