New results on the Zarankiewicz problem

Let image denote a bipartite graph with classes X and Y such that image and image. A complete bipartite subgraph with s vertices in X and t vertices in Y is denoted by image. The Zarankiewicz problem consists in finding the maximum number of edges, denoted by image, of a bipartite graph image with image and image without a complete bipartite image as a subgraph. First, we prove that image if image. Then we characterize the family image of extremal graphs for the values of parameters described above. Finally, we study the image case. We give the exact value of image if image and we characterize the extremal graphs if either image or both image and image.