Inequalities for two systems of subspaces with prescribed intersections

Let $W$ denote a linear space over a fixed field ${\mathbb F}$. We define the notions of weak $ISP$-system and weak $(u,v)$-system $\cal S=\{(U_i,V_i):~ 1\leq i\leq m\}$ of subspaces of $W$. We give upper bounds for the size of weak $ISP$-systems and weak $(u,v)$-systems.