Geometrical solution of an intersection problem for two hypergraphs (Q798334)

The author generalizes the following known theorem. If \(A_ 1,...,A_ m\) are a-element and \(B_ 1,...,B_ m\) are b-element sets with \(A_ i\cap B_ j=\emptyset\) iff \(i=j\) then \(m\leq \left( \begin{matrix} a+b\\ a\end{matrix} \right).\) He proves two generalizations below. Let \(A_ 1,...,A_ m\) be a-element and \(B_ 1,...,B_ m\) be b-element sets and \(t\leq\min (a,b).\) If \(| A_ i\cap B_ j|\leq t\) iff \(i=j\) then \(m\leq \left( \begin{matrix} a+b-2t\\ a-t\end{matrix} \right).\) The second result states that if \(A_ 1,...,A_ m\) are a-dimensional and \(B_ 1,...,B_ m\) are b-dimensional subspaces of \(R^ n\), and \(t\leq\min (a,b)\) then \(m\leq \left( \begin{matrix} a+b-2t\\ a-t\end{matrix} \right)\) provided \(\dim (A_ i\cap B_ j)\leq t\) iff \(i=j\).