This is the [Zarankiewicz problem][1].  It has also appeared on Mathoverflow several times, for example [here][2].

  [1]: https://en.wikipedia.org/wiki/Zarankiewicz_problem
  [2]: https://mathoverflow.net/questions/251959/maximum-number-of-edges-in-bipartite-graph-without-cycles-of-length-4/251962#251962