Turan number of bipartite graphs with no Kt,t.

The extremal number of a graph H, denoted by ex(n,H), is the maximum number of edges in a graph on n vertices that does not contain H. The celebrated Kővári-Sós-Turán theorem says that for a complete bipartite graph with parts of size t ≤ s the extremal number is ex(Ks,t) = O(n2-1/t). It is also known that this bound is sharp if s > (t−1)!. In this paper, we prove that if H is a bipartite graph such that all vertices in one of its parts have degree at most t but H contains no copy of Kt,t, then ex(n,H) = o(n2-1/t). This verifies a conjecture of Conlon, Janzer, and Lee. [ABSTRACT FROM AUTHOR]