Uniform hypergraphs containing no grids

A hypergraph is called an r×rr×rgrid   if it is isomorphic to a pattern of rr horizontal and rr vertical lines, i.e., a family of sets {A1,…,Ar,B1,…,Br}{A1,…,Ar,B1,…,Br} such that Ai∩Aj=Bi∩Bj=0̸Ai∩Aj=Bi∩Bj=0̸ for 1≤i