Let \(X,Y\) be disjoint finite sets. The family \({\mathcal F}=\{(F,G)\} \subset 2^ X \times 2^ Y\) is \((s,t,u)\)-intersecting if every pair \((F,G)\), \((F',G') \in {\mathcal F}\) satisfies \(| F \cap F' | \geq s\), \(| G \cap G' | \geq t\), and \(| F \cap F' |+| G \cap G' | \geq u\). The paper generalizes a result of Sali and gives exact upper bound for the size of \((s,t,s+t+1)\)-intersecting families. The extreme families are in close connection with Katona's theorem on maximal \(s\)- intersecting families. The main tools of this paper are Matsumoto and Tokushige's version of the Kruskal-Katona theorem and a new weight function inequality on Kneser graphs. This last result seems to be interesting in its own right as well.