Uniform hypergraphs containing no grids

A hypergraph is called an r×r$r\times r$grid   if it is isomorphic to a pattern of r$r$ horizontal and r$r$ vertical lines, i.e., a family of sets {_A1,…,_Ar,_B1,…,_Br}$\{A_1,\dots ,A_r,B_1,\dots ,B_r\}$ such that _Ai∩_Aj=_Bi∩_Bj=0?$A_i\cap A_j=B_i\cap B_j=0?$ for 1≤i<j≤r$1\le i<j\le r$ and |_Ai∩_Bj|=1$|A_i\cap B_j|=1$ for 1≤i,j≤r$1\le i,j\le r$. Three sets _C1,_C2,_C3$C_1,C_2,C_3$ form a triangle   if they pairwise intersect in three distinct singletons, |_C1∩_C2|=|_C2∩_C3|=|_C3∩_C1|=1$|C_1\cap C_2|=|C_2\cap C_3|=|C_3\cap C_1|=1$, _C1∩_C2≠_C1∩_C3$C_1\cap C_2\ne C_1\cap C_3$. A hypergraph is linear  , if |E∩F|≤1$|E\cap F|\le 1$ holds for every pair of edges E≠F$E\ne F$.