The Zarankiewicz problem in 3-partite graphs

Let F be a graph, k ≥ 2 be an integer, and write exχ≤k(n, F ) for the maximum number of edges in an n-vertex graph that is k-partite and has no subgraph isomorphic to F . The function exχ≤2(n, F ) has been studied by many researchers. Finding exχ≤2(n,Ks,t) is a special case of the Zarankiewicz problem. We prove an analogue of the Kövári-Sós-Turán Theorem by showing exχ≤3(n,Ks,t) ≤ ( 1 3 )1−1/s( t− 1 2 + o(1) )1/s n2−1/s for 2 ≤ s ≤ t. Using Sidon sets constructed by Bose and Chowla, we prove that this upper bound in asymptotically best possible in the case that s = 2 and t ≥ 3 is odd, i.e., exχ≤3(n,K2,2t+1) = √ t 3n 3/2 + o(n3/2) for t ≥ 1.