Graphs with no induced $K_{2,t}$

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

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