  Consider a graph $G$ on $n$ vertices with $\alpha \binom{n}{2}$ edges which
does not contain an induced $K_{2, t}$ ($t \geqslant 2$). How large does
$\alpha$ have to be to ensure that $G$ contains, say, a large clique or some
fixed subgraph $H$? We give results for two regimes: for $\alpha$ bounded away
from zero and for $\alpha = o(1)$.
  Our results for $\alpha = o(1)$ are strongly related to the Induced Tur\'{a}n
numbers which were recently introduced by Loh, Tait, Timmons and Zhou. For
$\alpha$ bounded away from zero, our results can be seen as a generalisation of
a result of Gy\'{a}rf\'{a}s, Hubenko and Solymosi and more recently Holmsen
(whose argument inspired ours).
