Graphs with no induced K2,t

Consider a graph G on n vertices with alpha((n)(2)) edges which does not contain an induced K-2,K-t (t >= 2). How large must alpha 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 Turan 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 Gyarfas, Hubenko and Solymosi and more recently Holmsen (whose argument inspired ours).