On two set-systems with restricted cross-intersections (Q1902973)

Let \(A_1,\dots, A_m\) and \(B_1,\dots, B_m\) be finite sets and \(r\), \(s\), \(l\), \(t\) be given numbers such that \(l\geq t\geq 0\) and \(|A_i|\leq r\), \(|B|\leq s\), \(|A_i\cap B_i|\leq t\) (for \(1\leq i\leq m)\) and \(|A_i\cap B_j|> l\) (for \(1\leq i< j\leq m\)). The author obtains an asymptotic formula for the maximal \(m= m(r, s, l, t)\) and some exact results under the assumption of the existence of a Sperner system. He also considers the corresponding problem for subspaces of a linear space over a finite field and obtains bounds for \(m\), which are attainable under similar conditions.