An extremal problem concerning graphs not containing Kt and Kt,t

We consider the following problem. Let t and n be positive integers with n⩾t⩾2. Determine the maximum number of edges of a graph of order n that contains neither Kt nor Kt,t as a subgraph. For n<2t, it is solved by Turánʹs theorem and for n=2t, by Brualdi and Mellendorf (Electron. J. Combinatorics 1 (1994) #R2). We solve it for n=2t+1.